Also available at http://amc.imfm.si
ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 5 (2012) 259–267
Duality on hypermaps with symmetric or
alternating monodromy group
Daniel Pinto ∗
CMUC, Department of Mathematics, University of Coimbra
3001-454 Coimbra, Portugal
Received 23 March 2011, accepted 23 January 2012, published online 22 March 2012
Abstract
Duality is the operation that interchanges hypervertices and hyperfaces on oriented hy-
permaps. The duality index measures how far a hypermap is from being self-dual. We say
that an oriented regular hypermap has duality-type {l, n} if l is the valency of its vertices
and n is the valency of its faces. Here, we study some properties of the duality index of
oriented regular hypermaps and we prove that for each pair n, l ∈ N, with n, l ≥ 2 (but not
both equal to 2), it is possible to find an oriented regular hypermap with extreme duality
index and of duality-type {l, n}, even if we are restricted to hypermaps with alternating or
symmetric monodromy group.
Keywords: Hypermaps, duality, primitive groups, alternating group, symmetric group, generators.
Math. Subj. Class.: 05C10, 05C25, 20B15, 20B35, 20F05
1 Introduction
One can look at a hypergraph as a generalization of the well known notion of a graph, since
an important restriction is removed: in a hypergraph, an (hyper)edge might connect more
than two (hyper)vertices. In a similar way, a hypermap is an obvious generalization of the
concept of a map. Although the word map is often used in mathematics with a different
meaning, here it is defined as a cellular embedding of a connected graph on a compact
connected surface. Without surprise, it is somehow natural to say that a hypermap is, in
its topological form, a cellular embedding of a connected hypergraph. Or, alternatively,
one can look at a hypermap as an embedding of a connected trivalent graph on a compact
connected surface, labelling each face 0, 1 or 2, so that each edge of the trivalent graph is in-
cident to two faces carrying different labels. In this representation of the hypermap (James
∗Research supported by CMUC and FCT (Portugal), through European program COMPETE/FEDER.
E-mail address: dpinto@mat.uc.pt (Daniel Pinto)
Copyright c© 2012 DMFA Slovenije
260 Ars Math. Contemp. 5 (2012) 259–267
representation [9]), the hypervertices are usually represented by label 0, the hyperedges by
label 1, and the hyperfaces by label 2. We should note that, by doing this, each hyperedge
of the hypermap might be adjacent to more than two hypervertices and, consequently, have
valency greater than 2 (something that does not occur in a map). If the valencies of all hy-
perfaces are equal, and the same happens with the hypervertices and with the hyperedges,
we say that the hypermap is uniform. Here, all the surfaces will be orientable and without
boundary. It follows that every hypermap can give rise to two oriented hypermaps, one for
each fixed orientation. If there is an automorphism that sends one of those oriented hyper-
maps into the other, we say that the hypermap is reflexible. Otherwise, we call it chiral.
The topological idea of an oriented hypermap (a cellular embedding of a hypergraph on an
oriented surface), briefly sketched here, has a very useful algebraic translation. In fact, one
can look at the results of this paper as purely algebraic results, although they certainly have
an important topological meaning.
A hypermap can be regarded as a transitive permutation representation ∆→ Sym Ω of
the group
∆ = 〈r0, r1, r2 | r20 = r21 = r22 = 1〉 ∼= C2 ∗ C2 ∗ C2,
on a set Ω representing its hyperflags (the cells of the barycentric subdivision of the hyper-
map). Similarly, an oriented hypermap (without boundary) can be regarded as a transitive
permutation representation of the subgroup
∆+ = 〈ρ0, ρ1, ρ2 | ρ0ρ1ρ2 = 1〉 = 〈ρ0, ρ2 | −〉
of index 2 in ∆ (a free group of rank 2) consisting of the elements of even word-length in
the generators ri, where ρ0 = r1r2, ρ1 = r2r0 and ρ2 = r0r1. In the case of hypermaps,
the hypervertices, hyperedges and hyperfaces (i-dimensional constituents for i = 0, 1, 2)
are the orbits of the dihedral subgroups 〈r1, r2〉, 〈r2, r0〉 and 〈r0, r1〉, and in the case of
oriented hypermaps they are the orbits of the cyclic subgroups 〈ρ0〉, 〈ρ1〉 and 〈ρ2〉, with
incidence given by nonempty intersection in each case. The local orientation around each
hypervertex, hyperedge or hyperface is determined by the cyclic order of the corresponding
cycle of ρ0, ρ1 or ρ2.
An oriented hypermap is regular if it has the highest degree of symmetry. It is well
known that, algebraically, an oriented regular hypermap H can be represented by a triple
(G, x, y), where the group G, a quotient of ∆+, is generated by the elements x and y. If
we want to look at those generators from a topological point of view, x can be interpreted
as the permutation that cyclically permutes the hyperdarts (oriented hyperedges) based on
the same hypervertex, and y the permutation that cyclically permutes the hyperdarts based
on the same hyperface, according to the chosen orientation. It is not difficult to verify that
every regular hypermap is uniform (in the James representation: every face labelled i, with
i ∈ {0, 1, 2}, has the same valency). We say that a regular hypermap has type (l,m, n) (for
l,m, n ∈ N) if l,m and n are the valencies of its hypervertices, hyperedges and hyperfaces,
respectively. The group G = 〈x, y〉 is called the monodromy group of the hypermap (and,
since the hypermap is regular, it coincides with its automorphism group). Duality, one
of the many possible operations on hypermaps (see [8]), is an operation that interchanges
hyperfaces and hypervertices, which in terms of the generators of the monodromy group
is the same as saying that the dual of H = (G, a, b) is Hd = (G, b, a). If there is an
automorphism of G that interchanges a and b, we say that H is self-dual (invariant under
duality) andH ∼= Hd.
D. Pinto: Duality on hypermaps with symmetric or alternating monodromy group 261
2 Duality index
A possible way to define the duality group of a hypermap H = (G, x, y), as an extension
of the notion of chirality group (see [1], [8]), is to say that it is the minimal subgroup
D(H) E G such that the quotient hypermap
H/D(H) = (G/D(H), xD(H), yD(H))
is a self-dual hypermap. The order of this duality group is called the duality index d of H,
and it is a measure of how far a hypermap is from being self-dual. If the duality index is
1, then the hypermap is self-dual; and the bigger that index, the more distant the hypermap
is from being self-dual. If D(H) ∼= G, the duality group is isomorphic to the monodromy
group of the hypermap and we say that the hypermap has extreme duality index.
3 The symmetric and the alternating groups
Before dealing with more difficult problems, we start this section by proving a simple
lemma:
Lemma 3.1. For every n ∈ N:
a) there is a non-self-dual hypermap with monodromy group Sn.
b) there is a non-self-dual hypermap with monodromy group An.
Proof. a) If n > 2, we take x = (1, 2, ... , n) and y = (1, 2). These permutations
generate the group Sn but, because they have different orders, there is no automorphism
that interchanges those two generators. It follows that the hypermap H = (Sn, x, y) is not
self-dual. If n = 2, S2 ∼= C2 = 〈g〉 and (C2, 1, g) is not self-dual (if we take 1 as the
neutral element of C2).
b) For n > 3, we just need to apply that same idea but using the canonical generators
x = (1, 2, ..., n) and y = (1, 2, 3), if n is odd, or x = (2, 3, .., n) and y = (1, 2, 3), if n is
even. Because, in both cases, the two generators do not have the same order, the hypermap
is not self-dual. If n = 3, An = A3 ∼= C3 = 〈g〉 and (C3, 1, g) is not self-dual.
Example 3.2. Let G = A5 = 〈x, y〉, where x = (1, 2, 3, 4, 5) and y = (1, 2, 3). Because
A5 is simple and the hypermap (A5, x, y) is not self-dual, the duality group (which is
normal in A5) must be A5 itself. Hence, the duality index of the hypermap is |A5| = 5!/2.
More generally, if we take
(An, (1, 2, ..., n), (1, 2, 3)), if n is odd, or (An, (2, 3, ..., n), (1, 2, 3)), if n is even,
we get hypermaps with extreme duality index and with monodromy group An. Hence:
Theorem 3.3. If n ≥ 3 there is a hypermap with extreme duality index and with mon-
odromy group An. 
262 Ars Math. Contemp. 5 (2012) 259–267
Remarks: a) Since each one of those hypermaps H has extreme duality index, D(H) =
Mon(H) = An. It follows that, for any n ≥ 3, An is the duality group of some hypermap.
b) If we take any simple group generated by two elements of different orders, we can get a
similar result.
If G = Sn, the only possible duality groups are 1, An and Sn. Therefore, a hypermap
with monodromy group Sn is self-dual, has extreme duality index or has n!/2 as duality
index. For n 6= 6, all automorphisms of Sn are inner and act by conjugation. If there is
an automorphism that transposes the two generators, the hypermap is self-dual and then
D(H) = 1. Otherwise, D(H) = An or Sn and we need to check if the hypermap with
monodromy group Sn/An, of order 2, is self-dual or not.
Theorem 3.4. Every hypermapH = (Sn, x, y):
i) has extreme duality index if x or y is an even permutation;
ii) is self-dual or has duality index n!/2 if x and y are both odd permutations and n 6= 4.
iii) is self dual or has duality index 4 if x and y are both odd permutations and n = 4.
Proof. i) The only non-identity quotients Sn/N of Sn are Sn/An ∼= C2, and S4/V4 ∼= S3
when n = 4. In each case, because one of xN and yN is in the unique subgroup of index 2
of Sn/N and the other is not, there can be no automorphism of Sn/N transposing xN and
yN . So, the only self-dual quotient is the trivial one and the hypermap has extreme duality
index.
ii) Sn is not cyclic and, by definition 〈x, y〉 = Sn. Hence, x 6= y. Suppose H is not
self-dual. Because H/An = (Sn/An, xAn, yAn) and |Sn/An| = 2, the two generators
xAn and yAn must be the same. It follows that the hypermap H/An is self-dual and
|D(H)| = n!/2.
iii) Let x and y be two odd permutations generating S4 and N the Klein group V4 (a
normal subgroup in S4). That pair of generators may be formed by a transposition and a
4-cycle or by two 4-cycles. Say x is a transposition and y is a 4-cycle. Then, they map
to distinct permutations in S4/N ∼= S3. Because the third element in S3 conjugates each
to the other, the quotient hypermap is self-dual (whereas the hypermap itself is not), with
duality group V4. If x and y are both 4-cycles the hypermap is self-dual.
Can we have an oriented regular hypermap with extreme duality index for each
type (l,m, n)?
The answer is no. There are no hypermaps of extreme duality index and of type (3, 2, 3),
since this has to be the tetrahedron, which is self-dual. On the other hand, if we restrict
ourselves to hyperbolic triples, where 1l +
1
m +
1
n < 1, we can enumerate orientably regular
hypermaps of a given type (l,m, n) with automorphism groups isomorphic to PSL(2, q)
or PGL(2, q) (this enumeration can be found in a joint work of Marston Conder, Primoz
Potocnik and Josef Siran [6], based on a paper by Sah [13]). Because these groups are
simple or almost simple, we can use them to try to find hypermaps with extreme duality
index or self-dual hypermaps. If l 6= n the hypermap cannot be self-dual. If l = n, we have
to check if there is an automorphism of PSL(2, q) or PGL(2, q) that interchanges the two
generators.
D. Pinto: Duality on hypermaps with symmetric or alternating monodromy group 263
We should also notice that for some triples (l,m, n) is very easy to find hypermaps with
extreme duality index and of that type.
If gcd(l, n) = 1 then
Gl,m,n = 〈a, b|al = bn = (ab)m = 1〉
is obviously the monodromy group of a hypermap of type (l,m, n). Let N be the duality
group of the hypermap (Gl,m,n, a, b). Then N is the smallest normal subgroup of Gl,m,n
such thatGl,m,n/N is reflexible, which means that we are working with the smallest normal
subgroup N of Gl,m,n such that the assignment a 7→ b, b 7→ a induces an automorphism
of Gl,m,n/N . We obtain this quotient by adding extra relations, substituting a for b and b
for a in the original ones. Then:
Gl,m,n/N = 〈a, b|al = an = bn = bl = (ab)m = (ba)m = 1〉.
Because l and n are co-prime, the group Gl,m,n/N collapses to the identity. Therefore,
H = (Gl,m,n, a, b) is a hypermap with extreme duality index and of type (l,m, n). This
proves that, if gcd(l, n) = 1, for every triple (l,m, n) there is a hypermap with extreme
duality index and of that type (l,m, n).
3.1 Alternating monodromy group
In the 1960’s, Graham Higman conjectured that any Fuchsian group has among its homo-
morphic images all but finitely many of the alternating groups. He also proved that An is
a factor group of (2, 3, 7) = 〈a, b|a2, b3, (ab)7〉 for all large n. Because 2, 3 and 7 are dis-
tinct prime numbers, we can conclude, from that result, that there is an infinite number of
hypermaps with extreme duality index and of type (2, 3, 7). The result obtained by Higman
was later extended by others (Marston Conder [5], for instance) that proved that the same
can be said for other families of triangle groups. The complete proof of the conjecture was
published in 2000 by Brent Everitt [7]. In his paper, it is shown that we need only consider
the triangle groups (p, q, r), 3 ≤ p < q < r, to prove the main result (which is done with
the help of coset diagrams for those triangle groups). Hence, it is possible to say that if
l, m, n are prime and l 6= n we can always find infinite hypermaps with extreme duality
index and of type (l,m, n), with alternating group as monodromy group. If p, q, r are not
all prime, then the alternating groups, being factor groups of the triangle group (p, q, r),
might correspond to hypermaps of type (p′, q′, r′) with p′|p, q′|q and r′|r, and not always
of type (p, q, r). It follows that we cannot directly apply the result of Brent Everitt to prove
that there are hypermaps of any type with alternating monodromy group. The proof of
Brent Everitt would had to be changed, in order to sustain that conclusion. Instead, we take
a different approach.
3.2 Duality-type of hypermaps with symmetric or alternating monodromy group
Definition 3.5. We say that an oriented regular hypermap has duality-type {l, n} if l is the
valency of its vertices and n is the valency of its faces (which is the same as saying that l
and n are the orders of the generators interchanged by the duality operation).
If we restrict ourselves to the family of hypermaps whose monodromy group is the
alternating or the symmetric group, can we have a hypermap with extreme duality index and
264 Ars Math. Contemp. 5 (2012) 259–267
of any duality-type {l, n}? In fact, we can not only prove that the answer is affirmative (if
n, l ≥ 2 but not both equal to 2) but also explicitly show how to construct such hypermaps.
The reason why, here, we look for duality-type instead of type is because, in the first case,
we just need to control the order of the two generators x and y, while in the second case
we also need to pay attention to the order of xy, which seems a harder problem to solve.
To complete that easier first task, we still need to add a few concepts and theorems about
primitive groups.
Let (G,Ω) be a permutation group. An equivalence relation ∼ is called G-invariant
if whenever α, β ∈ Ω satisfy α ∼ β then g(α) ∼ g(β) for all α, β ∈ Ω. Two obvious
G-invariant equivalence relations are: (i) α ∼ β if and only if α = β and (ii) α ∼ β for all
α, β ∈ Ω. We call (G,Ω) imprimitive if it admits some equivalence relation other than (i)
or (ii). Otherwise, we call (G,Ω) primitive.
There are several examples of primitive groups (Colva M. Roney-Dougal [11] has clas-
sified all the primitive permutation groups of degree less than 2500). For instance, any al-
ternating group An is primitive, and so is PGL(2, q), in its standard action, for any prime
power q. However, it is known - as underlined by Peter Cameron in the Encyclopedia of
Design Theory [3] - that primitive groups are “rare (for almost all n, the only primitive
groups of degree n are the symmetric and alternating groups, see [4]); and small (of or-
der at most nc·logn with known exceptions)”. The fact that most of the primitive groups
are alternating groups (which are simple) and symmetric groups (which have only one non
trivial normal subgroup) makes primitivity a powerful concept to build hypermaps with ex-
treme duality index. Although the probability of a primitive group being an alternating or
a symmetric group is very high, we need to be sure that we are not dealing with a different
kind of group. The next definitions and theorems help us to achieve that goal.
Definition 3.6. Let G be a permutation group on Ω and k a natural number with 1 ≤ k ≤
n = |Ω|. G is called k-transitive if for every two ordered k-tuples α1, ...αk and β1, ..., βk
of points of Ω (with αi 6= αj , βi 6= βj , for i 6= j) there exists g ∈ G which takes αt into
βt (for t = 1, ..., k).
Definition 3.7. The degree of a permutation group is the number of points moved by at
least one of its permutations.
Definition 3.8. The minimal degree of a permutation group is the minimum number of
points moved by any non-identity element of the group.
Theorem 3.9. [10] The minimal degree of a primitive group which is neither alternating
nor symmetric must exceed 4 whenever its degree exceeds 8.
Theorem 3.10. [14] A primitive group of degree n, which contains a cycle of degree m
with 1 < m < n is (n−m+ 1)-transitive.
Theorem 3.11. [12] A 2-transitive permutation group is primitive.
And, as a consequence of the classification of simple groups [2]:
Theorem 3.12. If a permutation group G is at least 6-transitive then G is the alternating
group or the symmetric group.
D. Pinto: Duality on hypermaps with symmetric or alternating monodromy group 265
With these tools we can now prove the following theorem:
Theorem 3.13. For each pair n, l ∈ N, with n, l ≥ 2 (but not both equal to 2), it is possible
to find an oriented regular hypermap with extreme duality index and of duality-type {l, n},
with alternating or symmetric group as monodromy group.
Proof. The general idea behind this proof is the following: first, we try to prove that the two
chosen permutations generate a primitive group, and then we show that they must generate
the symmetric or the alternating group. If they generate the symmetric group (which is
not a simple group), we also need to check that they give rise to a hypermap with extreme
duality index.
a) If l, n are not both even and l 6= n (we may assume, without loss of generality, that
l > n), let x = (1, 2, ..., l) and y = (1, 2, ..., n) be two permutations of Sl.
Then, since the stabilizer of the point l contains y and its conjugates under suitable
powers of x, the group G = 〈x, y〉 is 2-transitive. Hence, by Theorem 3.11, G is
primitive.
If we now take z as the commutator y−1x−1yx = (n, n− 1, l), we can, by Theorem
3.9, conclude that if l > 8, G must be the alternating or the symmetric group. If l
and n are both odd, G must be the alternating group and H = (G = Al, x, y) is a
hypermap of extreme duality index and of duality type {l, n}. If l is even and n is odd
(or if l is odd and n is even), G = 〈x, y〉 = Sl. Because not both of the generators
are odd permutations,H = (G, x, y) has extreme duality index (see Theorem 3.4).
(For l < 8 we can easily find two permutations, one of order l and another of order
n, that generate Al, when both permutations are odd, and Sl, in the other cases).
b) Suppose n = l (both odd).
Let the generators of the group be these two even permutations in Sl+1:
x = (1, 2, ..., l)
y = (2, 3, ..., l + 1).
The group generated by these two elements is primitive since it is a 2-transitive group
(Theorem 3.11). Moreover, x−1y−1xy = (1, l + 1)(l − 1, l) is a permutation that
just moves four points. It follows that G = 〈x, y〉 is a group of minimal degree ≤ 4.
Hence, by Theorem 3.9, 〈x, y〉 = Al if l + 1 ≥ 8, i.e. l ≥ 7. Cases for small l are
easy to solve.
c) If l and n are both even, we assume (without loss of generality) that l ≥ n. We then
take:
x = (1, 2, ..., l)
y = (1, 2)(l, l + 1, l + 2, ..., l + n− 1)
permutations of Sl+n−1 and G = 〈x, y〉. G is clearly a transitive group. Because
xy = (1, 3, 4, ..., l−1, l, l+1, l+2, ..., l+n−1) we can conclude that the stabilizer
of the point 2 acts transitively on the remaining points. Therefore, G = 〈x, y〉 is
a 2-transitive group and, by Theorem 3.11, G is also a primitive group of degree
l + n− 1.
266 Ars Math. Contemp. 5 (2012) 259–267
Since x is a cycle of order l and l+n−1− l+1 = n, the group G is n− transitive
(Theorem 3.10). For n > 5, G must be the alternating or symmetric group (Theorem
3.12). If n = 4 or n = 2 the degree of y2 does not exceed four and the group must be
the alternating or the symmetric group (Theorem 3.9). Since x is an odd permutation,
G = Sl+n−1. Then H = (Sl+n−1, x, y) is a hypermap with extreme duality index
because y is an even permutation (see Theorem 3.4).
Note: Because two involutions must always generate a dihedral group, no alternating group
can be generated by two involutions, and the only symmetric group with that property is S3.
But if an oriented hypermap has monodromy group S3 and is generated by two involutions,
it must be self-dual. Therefore, it is not possible to find an oriented regular hypermap with
extreme duality index and of duality-type {2, 2} with alternating or symmetric group as
monodromy group.
With the proof of the previous theorem, we have not only shown that it is possible to find
the required hypermaps, but also to describe them, since the generators of the alternating or
symmetric groups are explicitly given for each case. We have also proved a slightly weaker
result:
Theorem 3.14. For any l, n ∈ N, with n, l ≥ 2, one can find two permutations x and y
(of orders l and n, respectively), such that x and y generate an alternating or a symmetric
group. 
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