Also available at http://amc-journal.eu
ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 6 (2013) 393–408
Sectional split extensions arising from
lifts of groups
Rok Požar
Inštitut za matematiko, fiziko in mehaniko, Jadranska 19, 1000 Ljubljana, Slovenia
and
Fakulteta za matematiko, naravoslovje in informacijske tehnolgije, Univerza na
Primorskem, Glagoljaška 8, 6000 Koper, Slovenia
Received 3 September 2012, accepted 4 April 2013, published online 9 September 2013
Abstract
Covering techniques have recently emerged as an effective tool used for classification
of several infinite families of connected symmetric graphs. One commonly encountered
technique is based on the concept of lifting groups of automorphisms along regular cover-
ing projections ℘ : X̃ → X . Efficient computational methods are known for regular covers
with cyclic or elementary abelian group of covering transformations CT(℘).
In this paper we consider the lifting problem with an additional condition on how a
group should lift: given a connected graph X and a group G of its automorphisms, find
all connected regular covering projections ℘ : X̃ → X along which G lifts as a sectional
split extension. By this we mean that there exists a complement G of CT(℘) within the
lifted group G̃ such that G has an orbit intersecting each fibre in at most one vertex. As an
application, all connected elementary abelian regular coverings of the complete graph K4
along which a cyclic group of order 4 lifts as a sectional split extension are constructed.
Keywords: Covering projection, graph, group extension, lifting automorphisms, voltage assignment.
Math. Subj. Class.: 05C50, 05E18, 20B40, 20B25, 20K35, 57M10
1 Introduction
Graph covers play a significant role when symmetry properties of graphs are investigated.
One of the commonly used techniques is based on the concept of lifting automorphisms
along regular covering projections. Applications of this technique have been used to clas-
sify families of graphs with given structural properties (see for instance [2, 11, 12, 19, 20]).
E-mail address: pozar.rok@gmail.com (Rok Požar)
Copyright cchar∞3 2013 DMFA Slovenije
394 Ars Math. Contemp. 6 (2013) 393–408
In its most general form the problem of lifting automorphisms is well understood. Much
attention has been devoted to finding the necessary and sufficient lifting conditions in com-
binatorial terms, see [15, 16, 26, 27]. Nevertheless, these general results are rather hopeless
to apply when concrete examples and more detailed questions related to symmetry proper-
ties of graphs are considered.
In a more specific setting of regular covers in which the group of covering transforma-
tions is either cyclic or elementary abelian, the situation changes. For such covers, efficient
computational methods are known. For example, in the case of elementary abelian regu-
lar covers, the idea behind the approach developed in [19] is to reduce the general lifting
problem to that of finding invariant subspaces of matrix groups over prime fields, linearly
representing the action of automorphisms on the first homology group of the graph. Ap-
plying this method to a number of symmetric graphs – including the complete graphs K4
[20] and K5 [13], the Möbius-Kantor graph [18], the complete bipartite graph K3,3 [20],
the Petersen graph [21], the Pappus graph [25], the octahedron graph [14], and the Hea-
wood graph [19] – has resulted in the classification of connected elementary abelian regular
covers admitting various types of subgroups of automorphisms. A similar approach, also
based on linear criteria for lifting automorphisms, was proposed in [3], and has been used
in order to find connected regular coverings with cyclic or elementary abelian group of
covering transformation for the complete graph K4 [6], the 3-dimensional cube graph Q3
[7], the complete bipartite graph K3,3 [4], and the Petersen graph [5].
Assuming that a group G of automorphisms of X lifts along a regular covering projec-
tion ℘ : X̃ → X , the lifted group G̃ is an extension of the group of covering transformations
CT(℘) by G. Specific types of extensions have usually a strong impact on structural prop-
erties of the covering graph X̃ . In this context, the following two cases deserve special
attention: (i) G̃ is a split extension of CT(℘) by G, and in particular, (ii) G̃ is a direct split
extension of CT(℘) by G. For short we say that G lifts as a split extension or as a direct
split extension, respectively. In the former case there exists, by definition, a complement
G of CT(℘) within G̃, and a normal complement G of CT(℘) in the latter. This allows
us to compare actions of two isomorphic groups, G on X and G on X̃ , where G projects
isomorphically onto G along ℘. However, it can happen that the complement is not unique,
and what is more, different complements can exhibit different actions on X̃ . Therefore, the
analysis can be quite complicated. Certain algorithmic aspects related to the question of
how difficult is to test conditions (i) and (ii) are considered in [22].
According to particular kinds of actions that can arise from complements, two extremal
cases seem to stand out: (iii) there exists a complement G that acts transitively on the
covering graph X̃ , and (iv), there exists a complement G that is sectional. By this we mean
that there is a section of X̃ – a set of vertices containing at most one vertex from each fibre
– invariant under the action ofG. For short we say thatG lifts as a transitive split extension
or as a sectional split extension, respectively. Clearly, one might further restrict conditions
(iii) and (iv) to normal complements. Certain particular questions along these lines have
been addressed in [1, 8, 16, 17].
Motivated by the above discussion, the following problem is of interest. Given a con-
nected graph X and a group G of its automorphisms, find all connected regular covering
projections ℘ : X̃ → X along which G lifts in a prescribed way. In this paper we restrict
to case (iv) – we introduce a method for finding regular coverings along which G lifts as a
sectional split extension.
The basic idea behind our approach is the following. First, we take the cone X̂ over
R. Požar: Sectional split extensions arising from lifts of groups 395
the graph X obtained by adding a new vertex ∗ joined to every vertex of X , together
with the group of automorphisms Ĝ of X̂ that fixes ∗ and acts on X as the group G.
Next, the condition for lifting G as a sectional split extension is reduced to the general
lifting problem of finding regular coverings of X̂ admitting the lift of Ĝ. Consequently,
the original problem can be solved as soon as the general lifting problem can be solved.
Our approach is illustrated on a concrete example: we construct all connected elementary
abelian regular coverings of the complete graph K4 along which a cyclic group of order 4
lifts as a sectional split extension.
The rest of the paper is organized as follows. In Section 2 we review some preliminary
concepts about regular graph covers and lifting automorphisms. In Section 3 we devise a
method for constructing connected regular covering projections along which G lifts as a
sectional split extension. A detailed example is provided in Section 4.
2 Preliminaries
A graph is an ordered quadruple X = (D,V ; beg,−1 ), where DX = D and VX = V are
disjoint sets of darts and vertices, respectively, beg is a mapping that assigns to each dart x
its initial vertex beg(x), and −1 is an involution interchanging every dart x and its inverse
dart x−1. For a dart x, its terminal vertex is the vertex end(x) = beg(x−1). The orbits of
−1 are called edges. An edge e = {x, x−1} is called a link whenever beg(x) 6= end(x). If
beg(x) = end(x), then the respective edge is either a loop or a semi-edge, depending on
whether x 6= x−1 or x = x−1, respectively. All graphs in this paper are assumed to be
finite, meaning that the sets of vertices and darts are finite.
A graph homomorphism f : Y → X is an adjacency preserving mapping taking darts
to darts and vertices to vertices, or more precisely, f(beg(x)) = beg(f(x)) and f(x−1) =
f(x)−1. An isomorphism is a bijective homomorphism. An isomorphism of a graph onto
itself is an automorphism. All automorphisms of a graph X together with composition of
automorphisms constitute the automorphism group Aut(X).
A surjective homomorphism ℘ : X̃ → X is called a regular covering projection if there
exists a semi-regular subgroup S℘ of Aut(X̃) such that its vertex orbits and dart orbits
coincide with vertex fibres ℘−1(v), v ∈ VX , and dart fibres ℘−1(x), x ∈ DX , respectively.
Two regular covering projections ℘ : X̃ → X and ℘′ : X̃ ′ → X are isomorphic if there
exist an automorphism g of X and an isomorphism g̃ : X̃ → X̃ ′ such that the following
diagram
X̃
g̃−−−−→ X̃ ′
℘
y y℘′
X −−−−→
g
X
commutes. In particular, if g = id then ℘ and ℘′ are equivalent. If, in the above setting,
X̃ = X̃ ′ and ℘ = ℘′, then we say that g lifts along ℘ or that g̃ is a lift of g along ℘.
A group G ≤ Aut(X) lifts if all g ∈ G lift. The collection of all lifts of all elements
in G forms a subgroup G̃ ≤ Aut(X̃), the lift of G. In particular, the lift of the trivial
group is known as the group of covering transformations and denoted by CT(℘). Observe
that G̃ is an extension of CT(℘) by G. Furthermore, if G lifts along a given projection
℘, then it lifts along any covering projection equivalent to ℘. This allows us to study lifts
of automorphisms combinatorially in terms of voltage assignments, a concept that we are
396 Ars Math. Contemp. 6 (2013) 393–408
going to describe now.
Let X be a graph and let N be an (abstract) group, called the voltage group. Assign
to each dart x of X a voltage ζx ∈ N in such a way that ζx−1 = ζ−1x . Such a function
ζ : DX → N is called a voltage assignment on X . Further, construct the derived graph
Cov(ζ) with vertex set VX ×N and dart set DX ×N , where beg(x, n) = (beg(x), n) and
(x, n)−1 = (x−1, n ζx). The projection onto the first coordinate ℘ζ : Cov(ζ)→ X is then
the derived regular covering projection, where the required semi-regular subgroup S℘ζ of
Aut(Cov(ζ)) arises from the action of N on the second coordinate by left multiplication
on itself. Conversely, any regular covering projection ℘ : X̃ → X can be reconstructed by
a voltage assignment ζ on X such that the projection ℘ζ derived from ζ is equivalent to ℘.
Moreover, one can assume that the voltage assignment ζ is T -reduced for some arbitrarily
chosen spanning tree T of X , meaning that ζx = 1 for all darts x in T , see [9] for more
details.
Consider now a regular covering projection ℘ of connected graphs. Then we say that
℘ is connected. Further, the semi-regular group S℘ is equal to CT(℘), and the voltage
assignment ζ that reconstructs the projection ℘ is valued in the voltage group N ∼= CT(℘)
(viewed as an abstract group). Such a voltage assignment ζ is also called connected. It is
well known that ζ is connected if and only if each element of N appears as the voltage of
some closed walk. Furthermore, by the basic lifting lemma [15, 16], an automorphism g
of X lifts along ℘ζ if and only if each closed walk with trivial voltage is mapped by g to a
walk with trivial voltage.
Two assignments ζ and ζ ′ on X are equivalent whenever the respective derived regular
covering projections ℘ζ and ℘ζ′ are equivalent. Assuming that both assignments are con-
nected and valued inN , then they are equivalent if and only if there exists an automorphism
of N mapping ζW to ζ ′W for each closed walk W at u0 [27].
For a given connected graph X and subgroup G ≤ Aut(X), the problem of finding
regular covering projections ℘ along which G lifts is very difficult in general. However, in
the case of elementary abelian regular coverings ℘ – that is, when CT(℘) is isomorphic to
an elementary abelian group – the necessary and sufficient lifting condition can be stated
combinatorially by means of voltages as follows, see [19].
Let p be a prime. The first homology group H1(X;Zp) is generated by the (directed)
cycles of X and is isomorphic to the elementary abelian group Zrp, where r is the Betti
number of the graph X . The group H1(X;Zp) is usually viewed as a vector space over
Zp of dimension r. Since each automorphism α ∈ Aut(X) maps a cycle in X to a cycle
in X , there is a natural action of α on H1(X;Zp) which induces a linear transformation
α# of H1(X;Zp). Choose a spanning tree T of X and exactly one dart from each edge
{x, x−1} that is not contained in T . Then the sequence x1, x2, . . . , xr ∈ DX\DT of all
such darts naturally defines an (ordered) basis BT = {C1, C2, . . . , Cr} of H1(X;Zp),
where Ci is the cycle arising from the spanning tree T and the dart xi. Next, denote the
matrix representation of α# with respect to the basis BT by Mα ∈ Zr,rp . Thus, a subgroup
G ≤ Aut(X) induces a subgroup MG = {Mg | g ∈ G} ≤ GL(r,Zp). By M tG we denote
the dual group consisting of all transposes of matrices in MG.
Theorem 2.1. ([19, Proposition 6.3, Corollary 6.5]) With the notation above, let ζ : DX →
Zd,1p be a T -reduced voltage assignment onX , and letZ ∈ Zd,rp be the matrix with columns
ζx1 , ζx2 , . . . , ζxr .
If Z has rank d, then the derived graph Cov(ζ) is connected and the following hold:
R. Požar: Sectional split extensions arising from lifts of groups 397
(i) A group G ≤ Aut(X) lifts along ℘ζ : Cov(ζ)→ X if and only if the columns of Zt
form a basis of a M tG-invariant d-dimensional subspace S(ζ) of Zr,1p ∼= H1(X;Zp).
(ii) If ζ ′ : DX → Zd,1p is another voltage assignment on X satisfying the above condi-
tions, then ℘ζ′ is equivalent to ℘ζ if and only if S(ζ ′) = S(ζ). Moreover, ℘ζ′ is
isomorphic to ℘ζ if and only if there exists an automorphism α ∈ Aut(X) such that
the matrix M tα maps S(ζ ′) onto S(ζ).
By Theorem 2.1, we can find all pairwise nonequivalent connected elementary abelian
regular coverings of X along which G lifts – in terms of voltages – as follows. First find
a basis {u1, u2, . . . , ud} for each M tG-invariant subspace U of Zr,1p . Next, for each basis
{u1, u2, . . . , ud} consider a matrix Z with rows ut1, ut2, . . . , utd, and then define the voltage
assignment ζU : DX → Zd,1p , mapping dart xi to the i-th column of Z, i = 1, 2, . . . , r,
and mapping all darts of T to the trivial voltage. Observe that the choice of a spanning
tree together with a sequence x1, x2, . . . , xr as well as choosing a basis for an invariant
subspace is irrelevant as long as we consider regular coverings up to equivalence. Thus,
the problem of finding connected elementary abelian regular coverings along which a given
group of automorphisms lifts translates to a purely algebraic question of finding invariant
subspaces of finite linear groups.
In this context, let A ∈ Zn,np be an n×n matrix over a field Zp, acting as a linear trans-
formation on the column vector space Zn,1p . Next, let κA(x) = f1(x)n1f2(x)n2 · · · fk(x)nk
be the characteristic polynomial and mA(x) = f1(x)s1f2(x)s2 · · · fk(x)sk the minimal
polynomial of A where polynomials fi are pairwise distinct and irreducible over Zp. Then
Zn,1p can be written as a direct sum of the A-invariant subspaces
Fn,1 = Kerf1(A)s1 ⊕ Kerf2(A)s2 ⊕ · · · ⊕ Kerfk(A)sk .
Moreover, all A-invariant subspaces appear as direct sums of some A-invariant subspaces
of Kerfi(A)si .
As for finding common invariant subspaces of a finite linear group, we can often ex-
ploit Maschke’s theorem which states that if the characteristic of the field does not divide
the order of the group, then the representation is completely reducible. In this case one
essentially needs to find just the minimal common invariant subspaces. In particular, if the
order of the matrix A is not divisible by p, each A-invariant subspace of Zn,1p is a direct
sum of the minimal ones. For a more detailed description of finding invariant subspaces we
refer the reader to [10].
3 Sectional split extensions
We start by giving a more precise definition of a sectional split extension mentioned in
the Introduction. Let ℘ : X̃ → X be a regular covering projection of connected graphs,
and let Ω be a nonempty set of vertices of X . A section over Ω is a set of vertices Ω̄ of
X̃ containing exactly one vertex from each vertex fibre over Ω. Further, let G be a group
of automorphisms of X . Assuming that Ω is invariant under the action of G, we say that
G lifts along ℘ to G̃ as a sectional split extension over Ω if the following two conditions
are met: (a) G lifts along ℘ and (b) there exist a complement G to CT(℘) within G̃ and
a section Ω̄ over Ω that is invariant under the action of G. Such a complement is called
sectional over Ω. The necessary and sufficient conditions for G to lift as a sectional split
extension over Ω in terms of voltages were given by Malnič et al. This is summarized in
the following theorem.
398 Ars Math. Contemp. 6 (2013) 393–408
Theorem 3.1. ([16, Theorem 9.1, Theorem 9.3]) With the notation and assumptions above,
a group G lifts along ℘ as a sectional split extension over Ω if and only if ℘ can be recon-
structed by a voltage assignment ζ on X such that the following condition
ζW = 1⇒ ζgW = 1 (3.1)
holds for each automorphism g ∈ G and each walk W in X with both its endpoints in Ω .
Firstly, note that this theorem is an extended version of an old result of Biggs [1], retold
in a different language. Secondly, Malnič took this result further in [17], and used it to
sketch a method for testing whether G lifts along ℘ as a sectional split extension over Ω.
The approach is based on introducing a new vertex joined to every vertex of Ω, and then
converting condition (3.1) to the general lifting problem (but no proof is given). In order to
exploit this idea in another direction (see below), we introduce the following notation.
The cone X̂(Ω) over the graphX is the graph obtained by adding a new vertex ∗ joined
to every vertex of Ω. Assuming that Ω is invariant under the action of G, we denote by
Ĝ the group of automorphisms of X̂(Ω) that fixes ∗ and acts on X as the group G. Also,
for any voltage assignment ζ on X , we extend ζ to a voltage assignment ζ̂ on X̂(Ω) by
assigning the trivial voltage to the extra darts. More precisely,
ζ̂x =
{
ζx, x ∈ DX ;
1, x ∈ DX̂(Ω)\DX .
Conversely, for a voltage assignment ζ on X̂(Ω) being trivial on the set of extra darts we
denote by ζ̄ the restriction of ζ to X . Clearly, if ζ is not trivial on the set of extra darts,
then we can always find an equivalent assignment that is. For example, we may choose a
spanning tree T ∗ of X̂(Ω) such that all extra darts are included in T ∗, and then take an
equivalent T ∗-reduced voltage assignment. Moreover, the following holds.
Proposition 3.2. Let ζ and ζ ′ be two equivalent connected voltage assignments on X̂(Ω),
that are trivial on the set of extra darts DX̂(Ω)\DX . Then their restrictions ζ̄ and ζ̄ ′ to X
are also equivalent. Hence they are either both connected or both disconnected.
Proof. By definition of equivalence, there exists an isomorphism g̃ from the derived graph
Cov(ζ) to the derived graph Cov(ζ ′) such that ℘ζ = g̃℘ζ′ . Clearly, g̃ maps the vertex fibre
℘−1ζ (∗) to the vertex fibre ℘
−1
ζ′ (∗). Therefore, when restricting to X , the isomorphism g̃
induces an isomorphism from the derived graph Cov(ζ̄) to Cov(ζ̄ ′) that gives rise to an
equivalence of ζ̄ and ζ̄ ′. It is then obvious that isomorphic graphs are either both connected
or both disconnected, as required.
We are now ready to forge a link between connected regular coverings of X along
which G lifts as a sectional split extension over Ω, and connected regular coverings of
X̂(Ω) admitting the lift of Ĝ. For completeness, we explicitly record the following theorem
and provide the proof.
Theorem 3.3. Let ℘ : X̃ → X be a regular covering projection of connected graphs, and
let G be a group of automorphisms of X . Suppose that a nonempty subset Ω of vertices
of X is invariant under the action of G. Then the group G lifts along ℘ as a sectional split
extension over Ω if and only if ℘ can be reconstructed by a voltage assignment ζ onX such
that Ĝ lifts along the derived regular covering projection ℘ζ̂ : Cov(ζ̂ )→ X̂(Ω).
R. Požar: Sectional split extensions arising from lifts of groups 399
Proof. Suppose that G lifts along ℘ as a sectional split extension over Ω. By Theorem 3.1,
there exists a voltage assignment ζ on X that reconstructs ℘ and satisfies condition (3.1).
Extend ζ to a voltage assignment ζ̂. We will show that Ĝ lifts along the projection ℘ζ̂
derived from ζ̂. Let W ∗ be a closed walk at ∗ in X̂(Ω) with ζ̂W∗ = 1, and let g∗ ∈ Ĝ.
In view of the basic lifting lemma we need to show that ζ̂g∗W∗ = 1. Write W ∗ as a
concatenation W ∗ = W ∗1W
∗
2 . . .W
∗
k of closed walks at ∗ such that W ∗i = PiWiQ
−1
i ,
whereWi : ui → vi is a walk inX with both its endpoints ui and vi in Ω, while Pi : ∗ → ui
and Qi : ∗ → vi are walks of length 1, for i = 1, 2, . . . , k. Observe that ζW1ζW2 . . . ζWk =
1. Now choose a vertex u0 ∈ Ω. Let Ri : u0 → ui and Si : u0 → vi be walks with
ζRi = ζSi = 1, for i = 1, 2, . . . , k (note that such walks always exist). Then the product
of walks W =
∏k
i=1RiWiS
−1
i is a closed walk at u0 with ζW = ζW1ζW2 . . . ζWk = 1.
By condition (3.1) we have that ζgW = 1 as well as ζgRj = ζgSj = 1, for i = 1, 2, . . . , k.
Thus ζgW1ζgW2 . . . ζgWk = 1 implies that ζ̂g∗W∗ = 1, as required.
Conversely, suppose that ℘ is reconstructed by a voltage assignment ζ on X such that
Ĝ lifts along the covering projection ℘ζ̂ . By Theorem 3.1, it is sufficient to prove that ζ
satisfies condition (3.1). Consider a walk W : u → v in X with both its endpoints u and
v in Ω such that ζW = 1. Let P : ∗ → u and Q : ∗ → v be the (unique) walks of length
1 in X̂(Ω). Then the closed walk W ∗ = PWQ−1 at ∗ has voltage ζ̂W∗ = 1. By the
basic lifting lemma we have ζ̂g∗W∗ = 1 for any automorphism g∗ ∈ Ĝ. Hence ζgW = 1,
completing the proof.
Coming back to methods for testing whether G lifts along ℘ as a sectional split exten-
sion over Ω, one possibility would be to use the latter theorem. However, from compu-
tational point of view that would be inefficient, since one has to seek for an appropriate
voltage assignment that reconstructs the cover. For a more adequate approach to this prob-
lem we refer the reader to [23].
As already mentioned, Theorem 3.3 can be efficiently exploited in another direction:
given a connected graph X , a group G of its automorphisms, and a nonempty subset
Ω ⊆ VX invariant under the action of G, find, up to equivalence, all connected regular
coverings ℘ : X̃ → X along which G lifts as a sectional split extension over Ω. As a
first step towards this aim we need to find, in view of Proposition 3.1 and Theorem 3.3,
all pairwise nonequivalent connected regular coverings of X̂(Ω) along which the group Ĝ
lifts – combinatorially reconstructed in terms of voltage assignments ζ being trivial on the
set of extra darts. Although each ζ is connected – as it reconstructs a connected cover – its
restriction ζ̄ to X , however, might be disconnected. Thus, additional testing whether ζ̄ is
connected is required. These remarks are formally gathered in the following theorem.
Theorem 3.4. Let X be a connected graph and Ω a nonempty subset of vertices of X that
is invariant under the action of a group of automorphisms G ≤ Aut(X). Further, let ζ
be a voltage assignment on X̂(Ω) that is trivial on the set of extra darts DX̂(Ω)\DX and
gives rise to a connected regular covering projection along which the group Ĝ lifts. If the
restriction ζ̄ to X is connected, then G lifts along the derived regular covering projection
℘ζ̄ as a sectional split extension over Ω. Moreover, any connected regular covering of X
along which G lifts as a sectional split extension over Ω arises in this way.
Remark 3.5. Even if ζ and ζ ′ are two nonequivalent connected assignments on X̂(Ω) such
that their restrictions ζ̄ and ζ̄ ′ to X are connected, it still might happened that ζ̄ and ζ̄ ′ are
400 Ars Math. Contemp. 6 (2013) 393–408
equivalent. Thus, additional testing is needed.
Now we can more precisely summarize our approach. First, construct all voltage as-
signments ζ on X̂(Ω) giving rise to pairwise nonequivalent connected regular covering
projections along which Ĝ lifts. Next, consider their restrictions ζ̄ to X and remove the
disconnected ones. Finally, do further reduction to obtain all voltage assignments on X
giving rise to pairwise nonequivalent connected regular covering projections along which
G lifts as a sectional split extension over Ω.
4 Elementary abelian regular covers of K4
In light of the discussion in Section 3 we now give an example to illustrate our approach.
Let X = K4 be the complete graph on the vertex set VX = {1, 2, 3, 4}, and let Ω = VX .
Further, denote by g = (1234) ∈ Aut(X) the automorphism of X . We compute all volt-
age assignments on X giving rise to pairwise nonequivalent connected elementary abelian
regular covering projections along which the cyclic group G = 〈g〉 lifts as a sectional split
extension over Ω.
To start with, we need to find all voltage assignments on X̂(Ω) giving rise to pairwise
nonequivalent connected elementary abelian regular coverings along which the group Ĝ =
〈g∗〉 lifts. Let T ∗ be the spanning tree of X̂(Ω) consisting of all extra darts, and let
x1 = (1, 2), x2 = (2, 3), x3 = (3, 4), x4 = (4, 1), x5 = (2, 4), x6 = (3, 1)
denote the six cotree darts of X̂(Ω). Denote by BT ∗ = {~xi | 1 ≤ i ≤ 6} the ordered basis
of the vector space H1(X̂(Ω);Zp), where ~xi is the cycle arising from the spanning tree T ∗
and the dart xi. Next, in view of the remarks given in Preliminaries, let (g∗)# be the linear
transformation of H1(X̂(Ω);Zp) induced by the natural action of g∗ on H1(X̂(Ω);Zp),
and let Mg∗ ∈ Z6,6p be its matrix representation with respect to the basis BT ∗ . By compu-
tation we obtain that
A = M tg∗ =

0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
1 0 0 0 0 0
0 0 0 0 0 1
0 0 0 0 −1 0
 .
By Theorem 2.1, we need to find A-invariant subspaces of Z6,1p . However, note that every
elementary abelian regular Zd,1p -cover ofX is disconnected if the dimension d is higher that
the Betti number of X . Since the Betti number of X is three, it is therefore enough to find
all A-invariant subspaces of dimension at most three. These subspaces define T ∗-reduced
voltage assignments
ζ : DX̂(Ω) → Z
d,1
p , d = 1, 2, 3
on X̂(Ω) that give rise to pairwise nonequivalent connected regular coverings of X̂(Ω)
along which Ĝ lifts. In addition, as already explained in the previous section, their restric-
tions ζ̄ to X might still be disconnected as well as connected but equivalent.
In order to test whether the restriction ζ̄ to X stays connected, let T be the spanning
tree ofX consisting of the edges {1, 2}, {1, 3} and {1, 4}. Denote byC1, C2 andC3 cycles
R. Požar: Sectional split extensions arising from lifts of groups 401
arising from the spanning tree T and darts x2, x3 and x5, respectively. The connectedness
condition, relative to the ordered basis BT = {C1, C2, C3} of H1(X;Zp), translates to the
requirement that the voltages
ζ̄C1 = ζC1 = ζx1 + ζx2 + ζx6 ,
ζ̄C2 = ζC2 = ζx3 + ζx4 − ζx6 ,
ζ̄C3 = ζC3 = ζx1 + ζx4 + ζx5
generate the voltage group Zd,1p . As for the test of equivalence, let ζ and ζ ′ be two T ∗-
reduced voltage assignments on X̂(Ω) arising from two different d-dimensional subspaces
U and U ′ of Z6,1p , respectively. Suppose that their restrictions ζ̄ and ζ̄ ′ are connected. Then
ζ̄ and ζ̄ ′ are equivalent, in view of [27], if and only if there exists an automorphism of Zd,1p
mapping
ζC1 7→ ζ ′C1 , ζC2 7→ ζ
′
C2 , and ζC3 7→ ζ
′
C3 .
For the purpose of finding A-invariant subspaces, note that κA(x) = (x4 − 1)(x2 + 1)
is the characteristic polynomial of A, while
mA(x) = x
4 − 1
is its minimal polynomial. Further, observe that the factorization ofmA(x) into irreducible
factors over Zp depends on the congruence class of p modulo 4, namely
mA(x) =

(x− 1)(x+ 1)(x2 + 1), p ≡ 3 (mod 4);
(x− 1)(x+ 1)(x− i)(x+ i), p ≡ 1 (mod 4), i2 = −1;
(x− 1)4, p = 2.
Therefore the analysis splits into three cases.
Case p ≡ 3 (mod 4).
In this case the representation of the group 〈A〉 is completely reducible, by Maschke’s
theorem. The eigenvalues are 1 and −1, both of multiplicity 1. The respective eigenspaces
are LA(1) = 〈v1〉 and LA(−1) = 〈v2〉, where
v1 = (1, 1, 1, 1, 0, 0)
t and v2 = (1,−1, 1,−1, 0, 0)t.
The whole space splits into a direct sum of A-invariant subspaces
Z6,1p = LA(1)⊕ LA(−1)⊕ Ker(A2 + I).
It is obvious that the 1-dimensionalA-invariant subspaces are LA(1) and LA(−1). The
respective lists of voltages for the base homology cycles C1, C2, C3 in X are 2, 2, 2 for the
one arising from LA(1), and 0, 0, 0 for the one arising from LA(−1). Thus, only LA(1)
gives rise to a connected cover of X , while LA(−1) does not.
Since the 2-dimensional A-invariant subspace arising from the direct sum LA(1) ⊕
LA(−1) does not give a connected cover of X , all others are necessarily contained in
Ker(A2 + I). These subspaces are of the form 〈v,Av〉, for v ∈ Ker(A2 + I). There are
402 Ars Math. Contemp. 6 (2013) 393–408
p2 + 1 distinct subspaces. To check which of these give rise to connected covers of X ,
choose a basis of Ker(A2 + I), for instance
b1 = (1, 0,−1, 0, 0, 0)t,
b2 = (0, 1, 0,−1, 0, 0)t,
b3 = (0, 0, 0, 0, 1, 0)
t,
b4 = (0, 0, 0, 0, 0, 1)
t.
An arbitrary vector v ∈ Ker(A2 + I) is then of the form v = (a, b,−a,−b, c, d)t, for some
a, b, c, d ∈ Zp, while Av = (b,−a,−b, a, d,−c)t. For convenience we denote
Wa,b,c,d = 〈(a, b,−a,−b, c, d)t, (b,−a,−b, a, d,−c)t〉.
Checking for connectedness gives that
(a, b)t+(b,−a)t+(d,−c)t, (−a,−b)t+(−b, a)t−(d,−c)t and (a, b)t+(−b, a)t+(c, d)t
should generate Z2,1p . The condition is reduced to requiring that (a+b+d,−a+b−c)t and
(a−b+c, a+b+d)t are linearly independent in Z2,1p . Let x = a+b+d and y = a−b+c.
The vectors (x, y)t and (−y, x)t are linearly dependent if and only if x2 + y2 ≡ 0 (mod p)
Since p ≡ 3 (mod 4), we must have x ≡ 0 (mod p) and y ≡ 0 (mod p). Thus a disconnected
cover of X is obtained if and only if c = −a + b and d = −a − b; in this case Wa,b,c,d is
generated by
va,b = a(1, 0,−1, 0,−1,−1)t + b(0,−1, 0, 1,−1, 1)t and Ava,b.
Observe that any va,b is contained in 〈v1,0, Av1,0〉. Hence 〈va,b, Ava,b〉 = 〈v1,0, Av1,0〉
for all a, b ∈ Zp. This is therefore the only A-invariant 2-dimensional subspace giving rise
to a disconnected cover of X . As for the remaining subspaces, these are Wa,b,c,d where
(c, d) 6= (−a+b,−a−b). Furthermore, these subspaces all give rise to equivalent coverings
of X . Indeed. Choose one of these subspaces, say
W1,1,0,0 = 〈(1, 1,−1,−1, 0, 0)t, (1,−1,−1, 1, 0, 0)t〉.
Let ζ and ζ ′ be two assignments arising from Wa,b,c,d and W1,1,0,0, respectively. The base
homology cycles C1, C2, C3 in X have the following voltages
ζC1 = (a+ b+ d,−a+ b− c)t, ζ ′C1 = (2, 0)
t,
ζC2 = (−a− b− d, a− b+ c)t, ζ ′C2 = (−2, 0)
t,
ζC3 = (a− b+ c, a+ b+ d)t, ζ ′C3 = (0, 2)
t.
By computation one can check that there exists a matrix in GL(2,Zp) ∼= Aut(Z2,1p ) taking
ζC1 , ζC2 , ζC3 to ζ
′
C1
, ζ ′C2 , ζ
′
C3
, respectively, if and only if (c, d) 6= (−a + b,−a − b), and
the claim is proved. As a representative of the above 2-dimensional subspaces we take
W1,1,0,0.
Any 3-dimensionalA-invariant subspace giving rise to a connected cover ofX is equiv-
alent to the homological cover ofX . So it is enough to find one such a subspace, if it exists.
For instance, we may take the subspace LA(1)⊕W1,1,0,0, as the reader can easily check.
R. Požar: Sectional split extensions arising from lifts of groups 403
Case p ≡ 1 (mod 4).
The representation of the group 〈A〉 is again completely reducible, by Maschke’s theorem.
The matrix A is diagonalizable, having the diagonal form diagA(1,−1, i, i,−i,−i).
Clearly, the 1-dimensional eigenspaces LA(1) and LA(−1) are the same as before,
where only LA(1) gives rise to a connected cover of X . As for the eigenvalues i and −i
satisfying i2 ≡ −1 (mod p), the respective eigenspaces LA(i) = 〈ui, vi〉 and LA(−i) =
〈u−i, v−i〉 are 2-dimensional, where
ui = (1, i,−1,−i, 1, i)t, u−i = (1,−i,−1, i, 1,−i)t,
vi = (1, i,−1,−i, 0, 0)t, v−i = (1,−i,−1, i, 0, 0)t.
The 1-dimensional subspaces in LA(i) can be conveniently parametrized as
W∞(i) = 〈ui〉,
Ws(i) = 〈sui + vi〉 = 〈(s+ 1, (s+ 1)i,−(s+ 1),−(s+ 1)i, s, si)t〉, s ∈ Zp,
while those in LA(−i) can be parametrized as
W∞(−i) = 〈u−i〉,
Ws(−i) = 〈su−i + v−i〉 = 〈(s+ 1,−(s+ 1)i,−(s+ 1), (s+ 1)i, s,−si)t〉, s ∈ Zp.
The conditions for connectedness of covers ofX arising fromW∞(i),Ws(i),W∞(−i)
and Ws(−i) become i− 2 6≡ 0 (mod p), s(i− 2) 6≡ 1− i (mod p), −i− 2 6≡ 0 (mod p) and
s(−i − 2) 6≡ 1 + i (mod p), respectively. We need to consider subcases p 6= 5 and p = 5
separately.
Let p 6= 5. Then i,−i 6= 2, and there are (2p+ 1) 1-dimensional subspaces giving rise
to connected covers of X , namely the set
Wi = {Ws(i) | s ∈ (Zp\{(1− i)(i− 2)−1}) ∪ {∞}}
of p subspaces in LA(i), the set
W−i = {Ws(−i) | s ∈ (Zp\{(1 + i)(−i− 2)−1}) ∪ {∞}}
of p subspaces in LA(−i), and the subspace LA(1). However, all subspaces inWi give rise
to equivalent coverings of X . To show this, let ζ and ζ ′ be two assignments arising from
Ws(i) and W∞(i), respectively. By computation we have
ζC1 = (s+ 1)(1 + i) + si, ζ
′
C1 = 1 + 2i,
ζC2 = − (s+ 1)(1 + i)− si = −ζC1 , ζ ′C2 = − 1− 2i = −ζ
′
C1 ,
ζC3 = (s+ 1)(1− i) + s = −iζC1 , ζ ′C3 = 2− i = −iζ
′
C1 .
Clearly, there exists an automorphism of Zp taking ζC1 , ζC2 , ζC3 to ζ ′C1 , ζ
′
C2
, ζ ′C3 , respec-
tively, if and only if ζC1 6= 0. In fact, we do have ζC1 6= 0 since s 6= (1 − i)(i − 2)−1.
Similarly, all subspaces inW−i give rise to equivalent coverings of X . As a representative
inWi we choose W0(i), while inW−i we choose W0(−i). In fact, there are exactly three
pairwise nonequivalent connected coverings ofX , namely the one arising from LA(1), and
404 Ars Math. Contemp. 6 (2013) 393–408
the two coverings arising from W0(i) and W0(−i). The respective lists of voltages for the
base homology cycles C1, C2, C3 in X are 2, 2, 2 for the one arising from LA(1), while
1 + i,−1− i, 1− i and 1− i,−1 + i, 1 + i for the other two covers. The reader may check
that there is no automorphism of Zp taking any of these triples to any other.
Let p = 5. Then for each s ∈ Z5 the subspace Ws(2) gives rise to a connected cover
of X , while the subspace W∞(2) does not. On the other hand, for each s 6= 3 we obtain
a connected cover of X arising from Ws(3), and one connected cover of X arising from
W∞(3). Together with the cover of X arising from LA(1) we therefore have 2p+ 1 = 11
connected covers. If ζ denotes an assignment arising from Ws(2), then the base homology
cycles in X have voltages ζC1 = 3(s + 1) + 2s = 3, ζC2 = −3(s + 1) − 2s = −ζC1 ,
ζC3 = −(s+ 1) + s = −2ζC1 . It is obvious that the subspaces Ws(2), s ∈ Z5, give rise to
equivalent coverings ofX . As a representative we takeW0(2). Let now ζ be an assignment
arising from Ws(3), where s ∈ Zp and s 6= 3. Further, let ζ ′ denote an assignment arising
from W∞(3). Then we have
ζC1 = 4(s+ 1) + 3s = 2s− 1, ζ ′C1 = 2,
ζC2 = − 4(s+ 1)− 3s = −ζC1 , ζ ′C2 = − 2 = −ζ
′
C1 ,
ζC3 = − 2(s+ 1) + s = −3ζC1 , ζ ′C3 = − 1 = −3ζ
′
C1 .
Clearly, multiplication by s + 2 takes ζC1 , ζC2 , ζC3 to ζ
′
C1
, ζ ′C2 , ζ
′
C3
, respectively. As a
representative we take W0(3). The reader may check that LA(1),W0(2) and W0(3) give
rise to pairwise nonequivalent coverings of X .
Let us now consider the 2-dimensional subspaces. We shall need the following lemma.
Lemma 4.1. Let T ∗ be a spanning tree of X̂(Ω) such that all extra darts are included in T ∗,
and let the sequence x1, x2, . . . , xn contain exactly one dart from each edge not contained
in T ∗. Further, let U , U ′, W,W ′ be subspaces of Zn,1p such that U ∩W = {0} = U ′∩W ′,
and let ζU , ζU′ , ζW , ζW ′ , ζU⊕W , ζU′⊕W ′ denote T ∗-reduced voltage assignments on X̂(Ω),
where the voltages of darts xi arise from U , U ′, W,W ′, U ⊕W,U ′ ⊕W ′, respectively.
Suppose that all their restrictions to X are connected. If the restrictions of ζU and ζU′ are
equivalent and the restrictions of ζW and ζW ′ are equivalent, then the restrictions of ζU⊕W
and ζU′⊕W ′ are also equivalent.
Proof. Since U ∩ W = {0} = U ′ ∩ W ′ we may assume, up to equivalence of regular
covering projections, that
ζU⊕Wx =
[
ζUx
ζWx
]
and ζU
′⊕W ′
x =
[
ζU
′
x
ζW
′
x
]
,
for all darts x in X̂(Ω). Let r be the Betti number of X , and let C1, C2, . . . , Cr be an
ordered basis ofH1(X,Zp). Since the restrictions of ζU and ζU
′ are equivalent, there exists
an invertible matrix A mapping voltages ζUCi to voltages ζ
U′
Ci
, i = 1, 2, . . . , r. Similarly,
there exists an invertible matrix B mapping voltages ζWCi to voltages ζ
W ′
Ci
, i = 1, 2, . . . , r.
Then the matrix [
A
B
]
is invertible and clearly takes voltages ζU⊕WCi to voltages ζ
U′⊕W ′
Ci
. Hence the restrictions of
ζU⊕W and ζU′⊕W ′ to X are equivalent.
R. Požar: Sectional split extensions arising from lifts of groups 405
In order to test which 2-dimensional subspaces give rise to connected and possibly
equivalent coverings of X we only need to check, by Lemma 4.1, the subspaces LA(i),
LA(−i), and the following direct sums:
LA(1)⊕W0(i), LA(1)⊕W0(−i), W0(i)⊕W0(−i).
First note that the two subspaces LA(i) and LA(−i) give rise to disconnected covers of
X since each can be written as a direct sum of two 1-dimensional subspaces, of which one
gives rise to a disconnected cover of X . As for the remaining subspaces, they all give rise
to connected and pairwise nonequivalent coverings of X . We leave this to the reader.
Finally, any 3-dimensional subspace giving rise to a connected cover ofX is equivalent
to the homological cover over X . So it is enough to identify one such subspace. The
reader can check that LA(1) ⊕ W0(i) ⊕ W0(−i) satisfies the connectedness condition.
This completes the analysis when p is odd.
Case p = 2.
In this case the representation of the group 〈A〉 is not completely reducible. First we need
an appropriate Jordan basis for the matrix A. Observe that the respective Jordan form has
two elementary Jordan matrices, one of size 4 and one of size 2. By computation, a Jordan
basis is, say,
v1 = (1, 1, 1, 1, 0, 0)
t,
b1 = (0, 1, 0, 1, 0, 0)
t,
b3 = (0, 0, 1, 1, 0, 0)
t,
b4 = (0, 0, 0, 1, 0, 0)
t,
v2 = (0, 0, 0, 0, 1, 1)
t,
b2 = (0, 0, 0, 0, 1, 0)
t,
where v1 and v2 are the eigenvectors, the 4-dimensional cyclic subspace is spanned by
v1, b1, b3, b4, and the 2-dimensional one by v2, b2.
There are exactly three 1-dimensional A-invariant subspaces, all contained in the 2-
dimensional eigenspace LA(1), namely W∞(1) = 〈v1〉, W0(1) = 〈v2〉 and W1(1) =
〈(1, 1, 1, 1, 1, 1)t〉. Only the latter two give rise to connected covers of X . Moreover, both
also give rise to equivalent coverings of X . As a representative we choose, say, W1(1).
The resulting cover is the canonical double cover.
As for the 2-dimensional A-invariant subspaces, there are exactly seven of them. One
is the eigenspace LA(1) = 〈v1, v2〉. The other six subspaces arise from vectors u ∈
Ker(A − I)2 \ LA(1). Such a 2-dimensional subspace consists of the following vectors:
0, u, Au, u+Au. ClearlyAu 6= u (since u is not an eigenvector), andAu ∈ Ker(A−I)2 \
LA(1) (as A2u = Au implies Au = u). Therefore the elements in Ker(A− I)2 \LA(1) in
the same 2-dimensional subspace come in pairs. As the set Ker(A− I)2 \ LA(1) contains
exactly 12 nontrivial vectors, there are exactly six subspaces of this kind. These can be
explicitly represented as 〈v1, b1〉, 〈v1, u1〉, 〈v2, b2〉, 〈v2, u2〉, 〈v1 + v2, u3〉, 〈v1 + v2, u4〉,
406 Ars Math. Contemp. 6 (2013) 393–408
where
u1 = (0, 1, 0, 1, 1, 1)
t,
u1 = (1, 1, 1, 1, 1, 0)
t,
u3 = (1, 0, 1, 0, 0, 1)
t,
u4 = (1, 0, 1, 0, 1, 0)
t.
The reader can check that the subspaces giving rise to connected covers of X are pairwise
equivalent. As a representative we choose, say, 〈v2, b2〉.
Consider now the 3-dimensional A-invariant subspaces. It is enough to find just one (if
it exists) giving rise to a connected cover ofX (which is then equivalent to the homological
cover of X). However, the reader can check that all 3-dimensional subspaces give rise to
disconnected covers of X . To this end we only provide a basis for each of them. Note
that there are seven 3-dimensional subspaces in all. Indeed, three such subspaces exist in
Ker(A− I)2, namely
〈v1, v2, b1〉, 〈v1, v2, b2〉, and 〈v1, v2, u4〉,
each containing LA(1). The other four arise as cyclic subspaces of the Jordan chains
of length 3 (note that there are 16 chains in all, and A acts semi-regularly on the set of
vectors in Ker(A− I)3 \Ker(A− I)2 with four orbits of size 4). The respective bases are
{v1, b1, b3}, {v1, b1, u5}, {v1, u6, u7}, {v1, u6, u8}, where
u5 = (0, 0, 1, 1, 1, 1)
t,
u6 = (1, 0, 1, 0, 1, 1)
t,
u7 = (1, 0, 0, 1, 0, 1)
t,
u8 = (1, 0, 0, 1, 1, 0)
t.
This completes the analysis for p = 2.
Remark 4.2. In order to further reduce these coverings up to isomorphism we can follow
(ii) of Theorem 2.1. The possibility that the projections in Table 1 are isomorphic is con-
verted to checking the pairs in rows 2, 3, 4 and those in rows 7, 8, 9. The reader can check
that rows 3 and 4 give rise to isomorphic covers as well as rows 7 and 8.
Remark 4.3. Consider the automorphism h = (12) ofX . Clearly, g and h generate the full
automorphism group Aut(X). Let (h∗)# be the linear transformation of H1(X̂(Ω);Zp)
induced by the natural action of h∗ on H1(X̂(Ω);Zp), and let Mh∗ ∈ Z6,6p be its matrix
representation with respect to the basis BT ∗ . By computation we have that
M th∗ =

−1 0 0 0 0 0
0 0 0 0 0 −1
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 −1 0 0
0 −1 0 0 0 0
 .
It is now easy to check that among subspaces in Table 1 only W1(1) is also M th∗ -invariant.
Thus, the canonical double cover of X is the only covering along which the full automor-
phism group Aut(X) lifts as a sectional split extension over Ω.
R. Požar: Sectional split extensions arising from lifts of groups 407
Table 1: All voltage assignments on K4 giving rise to pairwise nonequivalent connected
elementary abelian regular covering projections along which the cyclic group 〈g〉 of auto-
morphisms of K4 lifts as a sectional split extension. Additionally, coverings in rows 3 and
4 are isomorphic, as well as those in rows 7 and 8.
n Inv. subsp. ζx1 ζx2 ζx3 ζx4 ζx5 ζx6 Condition
1 W1(1)
[
1
] [
1
] [
1
] [
1
] [
1
] [
1
]
p = 2
2 〈v1〉
[
1
] [
1
] [
1
] [
1
] [
0
] [
0
]
p 6= 2
3 〈vi〉
[
1
] [
i
] [
−1
] [
−i
] [
0
] [
0
] p ≡ 1 (mod 4),
i2 = −1
4 〈v−i〉
[
1
] [
−i
] [
−1
] [
i
] [
0
] [
0
] p ≡ 1 (mod 4),
i2 = −1
5 〈v2, b2〉
[
0
0
] [
0
0
] [
0
0
] [
0
0
] [
1
1
] [
1
0
]
p = 2
6 W1,1,0,0
[
1
1
] [
1
−1
] [
−1
−1
] [
−1
1
] [
0
0
] [
0
0
]
p ≡ 3 (mod 4)
7 〈v1, vi〉
[
1
1
] [
1
i
] [
1
−1
] [
1
−i
] [
0
0
] [
0
0
]
p ≡ 1 (mod 4),
i2 = −1
8 〈v1, v−i〉
[
1
1
] [
1
−i
] [
1
−1
] [
1
i
] [
0
0
] [
0
0
]
p ≡ 1 (mod 4),
i2 = −1
9 〈vi, v−i〉
[
1
1
] [
i
−i
] [
−1
−1
] [
−i
i
] [
0
0
] [
0
0
]
p ≡ 1 (mod 4),
i2 = −1
10 〈v1,W1,1,0,0〉
11
1
  11
−1
  1−1
−1
  1−1
1
 00
0
 00
0
 p ≡ 3 (mod 4)
11 〈v1, vi, v−i〉
11
1
  1i
−i
  1−1
−1
  1−i
i
 00
0
 00
0
 p ≡ 1 (mod 4),
i2 = −1
Acknowledgement. The author would like to thank Aleksander Malnič for enlightening
discussions, and both referees for their helpful suggestions that improved the presentation.
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