ARS MATHEMATICA CONTEMPORANEA
ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 14 (2018) 177-186
https://doi.org/10.26493/1855-3974.1298.937
(Also available at http://amc-journal.eu)
Every finite group has a normal bi-Cayley
graph*
Jin-Xin Zhou *
Mathematics, Beijing Jiaotong University, Beijing 100044, P.R. China
Received 19 January 2017, accepted 27 April 2017, published online 17 August 2017
Abstract
A graph r with a group H of automorphisms acting semiregularly on the vertices with two orbits is called a bi-Cayley graph over H. When H is a normal subgroup of Aut(r), we say that r is normal with respect to H. In this paper, we show that every finite group has a connected normal bi-Cayley graph. This improves a theorem by Arezoomand and Taeri and provides a positive answer to a question reported in the literature.
Keywords: Normal, bi-Cayley, Cartesian product. Math. Subj. Class.: 05C25, 20B25
1 Introduction
Throughout this paper, groups are assumed to be finite, and graphs are assumed to be finite, connected, simple and undirected. For the group-theoretic and graph-theoretic terminology not defined here we refer the reader to [5, 23].
Let G be a permutation group on a set Q and a e Q. Denote by Ga the stabilizer of a in G, that is, the subgroup of G fixing the point a. We say that G is semiregular on Q if Ga = 1 for every a e Q and regular if G is transitive and semiregular. It is well-known that a graph r is a Cayley graph if it has an automorphism group acting regularly on its vertex set (see [4, Lemma 16.3]). If we, instead, require that the graph r admits a group of automorphisms acting semiregularly on its vertex set with two orbits, then we obtain the so-called bi-Cayley graph.
Cayley graph is usually defined in the following way. Given a finite group G and an inverse closed subset S C G \ {1}, the Cayley graph Cay(G, S) on G with respect to S is a
*This work was supported by the National Natural Science Foundation of China (11671030) and the Fundamental Research Funds for the Central Universities (2015JBM110).
^The author is indebted to the anonymous referee for many valuable comments and constructive suggestions. E-mail address: jxzhou@bjtu.edu.cn (Jin-Xin Zhou)
©® This work is licensed under http://creativecommons.org/licenses/by/3.0/
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graph with vertex set G and edge set {{g, sg} | g G G, s G S}. For any g G G, R(g) is the permutation of G defined by R(g): x ^ xg for x G G. Set R(G) := {R(g) | g G G}. It is well-known that R(G) is a subgroup of Aut(Cay(G, S)). In 1981, Godsil [10] proved that the normalizer of R(G) in Aut(Cay(G, S)) is R(G) x Aut(G, S), where Aut(G, S) is the group of automorphisms of G fixing the set S set-wise. This result has been successfully used in characterizing various families of GRRs, namely, Cayley graphs Cay(G, S) such that R(G) = Aut(Cay(G, S)) (see, for example, [10, 11]). A Cayley graph Cay(G, S) is said to be normal if R(G) is normal in Aut(Cay(G, S)). This concept was introduced by Xu in [24], and for more results about normal Cayley graphs, we refer the reader to [8].
Similarly, every bi-Cayley graph admits the following concrete realization. Given a group H, let R, L and S be subsets of H such that R-1 = R, L-1 = L and R U L does not contain the identity element of H. The bi-Cayley graph over H relative to the triple (R, L, S), denoted by BiCay(H, R, L, S), is the graph having vertex set the union of the right part H0 = {h0 | h G H} and the left part H1 = {h1 | h G H}, and edge set the union of the right edges {{h0, g0} | gh-1 G R}, the left edges {{h1, g1} | gh-1 G L} and the spokes {{h0, g1} | gh-1 G S}. Let r = BiCay(H, R, L, S). For g G H, define a permutation BR(g) on the vertices of r by the rule
hfR(g) = (hg),, Vi G Z2,h G H.
Then BR(H) = {BR(g) | g G H} is a semiregular subgroup of Aut(r) which is isomorphic to H and has H0 and H1 as its two orbits. When BR(H) is normal in Aut(r), the bi-Cayley graph r = BiCay(H, R, L, S) will be called a normal bi-Cayley graph over H (see [3] or [27]).
Wang et al. in [22] determined the groups having a connected normal Cayley graph.
Proposition 1.1. Every finite group G has a normal Cayley graph unless G = C4 x C2 or
G = Qg x C2(r > 0).
Following up this result, Arezoomand and Taeri in [3] asked: Which finite groups have normal bi-Cayley graphs? They also gave a partial answer to this question by proving that every finite group G = Qg x C2 (r > 0) has at least one normal bi-Cayley graph. At the end of [3], the authors asked the following question:
Question 1.2 ([3, Question]). Is there any normal bi-Cayley graph over G = Qg x C2 for each r > 0?
We remark that for every finite group G = Qg x C2 (r > 0), the normal bi-Cayley graph over G constructed in the proof of [3, Theorem 5] is not of regular valency, and so is not vertex-transitive. So it is natural to ask the following question.
Question 1.3. Is there any vertex-transitive normal bi-Cayley graph over a finite group G?
In this paper, Questions 1.2 and 1.3 are answered in positive. The following is the main result of this paper.
Theorem 1.4. Every finite group has a vertex-transitive normal bi-Cayley graph.
To end this section we give some notation which is used in this paper. For a positive integer n, denote by Cn the cyclic group of order n, by Zn the ring of integers modulo n, by D2n the dihedral group of order 2n, and by Alt(n) and Sym(n) the alternating group
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and symmetric group of degree n, respectively. Denote by Q8 the quaternion group. For two groups M and N, N x M denotes a semidirect product of N by M. The identity element of a finite group G is denoted by 1.
For a finite, simple and undirected graph r, we use V(T), E(T) and Aut(r) to denote its vertex set, edge set and full automorphism group, respectively, and for any u, v G V(r), u ~ v means that u and v are adjacent. A graph r is said to be vertex-transitive if its full automorphism group Aut(r) acts transitively on its vertex set. For any subset B of V(r), the subgraph of r induced by B will be denoted by r[B].
2 Cartesian products
The Cartesian product X □ Y of graphs X and Y is a graph with vertex set V(X) x V(Y), and with vertices (u, x) and (v, y) being adjacent if and only if u = v and x ~ y in Y, or x = y and u ~ v in X.
A non-trivial graph X is prime if it is not isomorphic to a Cartesian product of two smaller graphs. The following proposition shows the uniqueness of the prime factor decomposition of connected graphs with respect to the Cartesian product.
Proposition 2.1 ([12, Theorem 6.6]). Every connected finite graph can be decomposed as a Cartesian product of prime graphs, uniquely up to isomorphism and the order of the factors.
Two non-trivial graphs are relatively prime (w.r.t. Cartesian product) if they have no non-trivial common factor. Now we consider the automorphisms of Cartesian product of graphs.
Proposition 2.2 ([12, Theorem 6.10]). Suppose $ is an automorphism of a connected graph r with prime factor decomposition r = r □ r2 □ • • • □ rk. Then there is a permutation n of {1,2,..., k} and isomorphisms fa: r^) ^ r for which
^^	xk ) = ($1(xn(1) ),	. . . , (xn(fc})).
Corollary 2.3 ([12, Corollary 6.12]). Let r be a connected graph with prime factor decomposition r = r1 □ r2 □ • • • □ rk. If r1, r2,..., rk are relatively prime, then Aut(r) = Aut(ri) x Aut(r2) x • • • x Aut(rk).
The following theorem provides a method of constructing normal bi-Cayley graphs.
Theorem 2.4. Let X be a connected normal bi-Cayley graph over a group H, and let Y be a connected normal Cayley graph over a group K. If X and Y are relatively prime, then X □ Y is also a normal bi-Cayley graph over the group H x K.
Proof. Assume that X and Y are relatively prime. By Corollary 2.3, Aut(X □ Y) = Aut(X) x Aut(Y). Since X is a connected normal bi-Cayley graph over H, one has BR(H) < Aut(X), and since Y is a connected normal Cayley graph over a group K, one has R(K) < Aut(Y). Then BR(H) x R(K) is a normal subgroup of Aut(X □ Y) = Aut(X) x Aut(Y). Note that BR(H) acts semiregularly on V(X) with two orbits, and R(K) acts regularly on V(Y). It follows that BR(H) x R(K) acts semiregularly on V(X) x V(Y) with two orbits, and thereby X □ Y is also a normal bi-Cayley graph over the group H x K.	□
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3 Normal bi-Cayley graphs over Q8 x C (r > 0)
In this section, we shall answer Question 1.2 in positive.
3.1 The n-dimensional hypercube
For n > 1, the n-dimensional hypercube, denoted by Qn, is the graph whose vertices are all the n-tuples of 0's and 1's with two n-tuples being adjacent if and only if they differ in exactly one place.
Let N = Cn be an elementary abelian 2-group of order 2n with a minimum generating set S = {si, s2, «3,..., sn}. By the definition of Qn, we have Cay(N, S) = Qn. For convenience of the statement, we assume that Qn = Cay(N, S). If n =1, then Q1 = K2 and so Aut(Q1) = N. In what follows, assume that n > 2. It is easy to observe that for any distinct sj, sj there is a unique 4-cycle in Qn passing through 1, sj, sj, where 1 is the identity element of N. So if a subgroup of Aut(Qn) fixes S pointwise, then it also fixes every vertex at distance 2 from 1. By the connectedness and vertex-transitivity of Qn, we have Aut(Q„)i acts faithfully on S. It follows that Aut(Qn)i < Sym(n). On the other hand, it is easy to see that each permutation on S induces an automorphism of N, and so Aut(N, S) = Sym(n). Since Aut(N, S) < Aut(Q„)i, one has Aut(Q„)i = Aut(N, S) = Sym(n). Consequently, we have Aut(Qn) = R(N) x Aut(N, S) = N x Sym(n) (see also [25, Lemma 1.1]).
Note that Qn is bipartite. Let Aut(Q„)* be the kernel of Aut(Qn) acting on the two partition sets of Qn. Let E = R(N) n Aut(Qn)*. Then E < Aut(Q„)* and E < R(N). It follows that E < Aut(Q„)*R(N) = Aut(Q„). Clearly, E acts semiregularly on V(Q„) with two orbits. Thus, we have the following lemma.
Lemma 3.1. Use the same notation as in the above three paragraphs. For any n > 1, Qn is a normal Cayley graph over N, and Qn is also a normal bi-Cayley graph over E.
3.2 The Möbius-Kantor graph
The Möbius-Kantor graph GP(8,3) is a graph with vertex set V = {i, | i G Z8} and edge set the union of the outer edges {{i,i+1} | i G Z8}, the inner edges {{i', (i+3)'} | i G Z8}, and the spokes {{i, i'} | i G Z8} (see Figure 1). Note that GP(8,3) is a bipartite graph with bipartition sets B1 = {1,3, 5, 7,0', 2', 4', 6'} and B2 = {0, 2,4,6,1', 3', 5', 7'}.
In [26], the edge-transitive groups of automorphisms of Aut(GP(8,3)) were determined. We first introduce the following automorphisms of GP(8, 3), represented as permutations on the vertex set V:
a	=	(1 3 5 7)(0 2 4 6)(1' 3' 5' 7')(0' 2' 4' 6'),
P	=	(0 1' 2)(o' 6' 3)(4 5' 6)(7 4' 2'),
Y	=	(1 1')(2 6')(3 3')(4 0')(5 5')(6 2')(7 7' )(0 4'),
S	=	(1 1')(2 4')(3 7')(4 2')(5 5')(6 0')(7 3' )(0 6').
By [26, Lemma 3.1], we have (a, P) = (a, a^} x (P) = Q8 x Z3, where Q8 is the quaternion group, and moreover, (a, P) < Aut(GP(8,3)). Clearly, (a, a^} = Q8 is the Sylow 2-subgroup of (a, P), so (a, a^} is characteristic in (a, P), and then it is normal in Aut(GP(8, 3)) because (a, P) < Aut(GP(8, 3)). For convenience of the statement, we let Q8 = (a, a^). It is easy to see that Q8 acts semiregularly on V with two orbits B1 and B2. Thus we have the following lemma.
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0
3
1 2 Figure 1: The Mobius-Kantor graph GP(8,3).
Lemma 3.2. GP(8, 3) is a normal bi-Cayley graph over Q8.
6
5
7
4
3.3 An answer to Question 1.2
Noting that GP(8, 3) is of girth 6, GP(8, 3) is prime. For each r > 1, it is easy to see that Qr = K2 □ K2 □ • • • □ K2. So, Qn and GP(8, 3) are relatively prime. Now combining
n times
together Lemmas 3.1 and 3.2 and Theorem 2.4, we can obtain the following theorem.
Theorem 3.3. For each r > 1, GP(8,3) x Qr is a vertex-transitive normal bi-Cayley graph over Q8 x N, where N = C£.
4 Proof of Theorem 1.4
The proof of Theorem 1.4 will be completed by the following lemmas. Let G be a group. A Cayley graph r = Cay(G, S) on G is said to be a graphical regular representation (or GRR for short) of G if Aut(r) = R(G).
Lemma 4.1. Let G be a group admitting a GRR r. Then r □ K2 is a normal bi-Cayley graph over the group G.
Proof. If K2 and r are relatively prime, then by Corollary 2.3, we have Aut(r □ K2) = Aut(r) x Aut(K2). Clearly, R(G) x {1} acts semiregularly on V(r □ K2) with two orbits, and R(G) x {1} < Aut(r □ K2), where 1 is the identity of Aut(K2). It follows that r □ K2 is a normal bi-Cayley graph over the group G.
Suppose that K2 is also a prime factor of r. Let r = ri □ K2 □ • • • □ K2 be such
m times
that r1 is coprime to K2. From Corollary 2.3 it follows that G = Aut(r) = Aut(r1) x Aut(K2 □ • • • □ K2). Since r is a GRR of G, one has m = 1, and therefore r □ K2 = ri □ K2 □ K2. Then G = Aut(ri) x Aut(K2 □ K2), and ri is a GRR of Aut(ri). By Lemma 3.1, K2 □ K2 is a normal bi-Cayley graph over C2, and by Theorem 2.4, r □ K2 is a normal bi-Cayley graph over Aut(ri) x C2 = G.	□
A group G is called generalized dicyclic group if it is non-abelian and has an abelian
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subgroup L of index 2 and an element b e G \ L of order 4 such that b-1xb = x-1 for every x e L.
The following theorem gives a list of groups having no GRR (see [9]).
Theorem 4.2. A finite group G admits a GRR unless G belongs to one of the following classes of groups:
(I) Class C: abelian groups of exponent greater than two; (II) Class D: the generalized dicyclic groups; (III) Class E: the following thirteen "exceptional groups":
(1)	Z2, Z3, Z2;
(2)	De,Ds,Dio;
(3)	A;
(4)	(a, b, c | a2 = b2 = c2 = 1, abc = bca = cab);
(5)	(a, b | a8 = b2 = 1,bab = a5);
(6)	(a, b, c | a3 = b3 = c2 = 1, ab = ba, (ac)2 = (cb)2 = 1);
(7)	(a, b, c | a3 = b3 = c3 = 1, ac = ca, bc = cb, c = a-1b-1 cb);
(8)	Q8 x Z3, Q8 x Z4.
Lemma 4.3. Let G be a group in Class D of Theorem 4.2. Then G has a normal bi-Cayley graph.
Proof. If G = Q8 x C2 for some r > 0, then by Theorem 3.3 and Lemma 3.2, G has a normal bi-Cayley graph. In what follows, we assume that G ^ Q8 x C2 for any r > 0. By Proposition 1.1, G has a normal Cayley graph, say r. If r is coprime to K2, then by Corollary 2.3, r □ K2 is a normal bi-Cayley graph over G.
Now suppose that K2 is a prime factor of r. Let r = r1 □ Qm, where Qm = K2 □ • • • □ K2 and r1 is coprime to K2. Again by Corollary 2.3, we have Aut(r) =
m times
Aut(r1) x Aut(Qm). For any x e V(Qm), set Vx = {(u, x) | u e V(r1)}, and for any y e V(r1), set Uy = {(y,v) | v e V(Qm)}. Then r[Vx] = r1 and r[Vy] = Qm. Let Gyx and GUy be the subgroups of G fixing Vx and Uy setwise, respectively. We shall prove the following claim.
Claim. G = GVx x GUy, r1 is a normal Cayley graph over a group which is isomorphic
to Gvx, and Guy = C^.
Since r is vertex-transitive, by Proposition 2.2, Vx is an orbit of Aut(r1) x {1} and Aut(r1) x {1} = Aut(r[Vx]). As Aut(r1) x {1} < Aut(r), each Vx is a block of imprim-itivity of Aut(r) (namely, either Vxg = Vx or Vxg n Vx = 0 for any g e Aut(r)). Consider the quotient graph r' with vertex set {Vx | x e V(Qm)}, and Vx is adjacent to Vx if and only if x is adjacent to x' in Qm. Then r' = Qm, and Aut(r1) x {1} is just the kernel of Aut(r) acting on V(r'). This implies that the subgroup Aut(r)Vx of Aut(r) fixing Vx set-wise is just Aut(r1) x Aut(Qm)x. Since G is regular on V(r), GVx is also regular on Vx, and so r1 = r[Vx] may be viewed as a Cayley graph on GVx. Since G < Aut(r), one has GVx = G n Aut(r)Vx < Aut(r)Vx. Note that {1} x Aut(Qm)x fixes every vertex in Vx. It follows that GVx n ({1} x Aut(Qm)x) is trivial, and so GVx can be viewed as a
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normal regular subgroup of Aut(Ti) x {1}. Therefore, r is a normal Cayley graph over some group, say H = GVx.
With a similar argument as above, we can show that Qm is also a normal Cayley graph over some group, say K = GUy. From the argument in Section 3.1, we have Aut(Qm) = N x Sym(m) with N = C". We claim that K = N .If this is not true, then we would have 1 = KN/N < Aut(Qm)/N = Sym(m), and since K is a 2-group, the only possibility is m = 4. However, by Magma [6], Aut(Q4) has only one normal regular subgroup which is isomorphic to C|, a contradiction. Thus, K = N = C", and hence GUy = C2m.
For any g G GV n GU , we have g fixes (y, x) and so g = 1 because G is regular on V (r). Thus, Gvx nGuy =y {1}. Then |Gyx Guy | = |Gyx ||Guy | = |V*||Uy | = |V (r)| = |G|. It follows that G = GVxGUy. To show that G = GVx x GUy, it suffices to show that both GVx and GUy are normal in G. As G is a generalized dicyclic group, it is non-abelian and has an abelian subgroup L of index 2 and an element b G G \ L of order 4 such that b-1ab = a-1 for every a G L.
Suppose that GUy ^ L. Then there exists g G GUy such that g = ab® for some a G L and i = 1 or -1. Since GUy = C", g is also an involution, and so G = L x (g). Clearly, for any a G L, we have g-1ag = a-1, and so (ga)2 = 1. This would force that every element of G outside L is an involution, a contradiction. Thus, GUy < L, and hence
GU < G. Uy
Since G = GVx GUy, GUy < L implies that GVx ^ L. Then |GVx : GVx n L| =2 since |G : L| = 2. It then follows that GVx n L < G, and hence
G/Gvx n L = (GUy (Gvx n L)/Gvx n L) x (Gyx/Gyx n L).
Again as G is a generalized dicyclic group and since GUy < L, the non-trivial element of GVx/GVx n L maps every element of GUy (GVx n L)/GVx n L to its inverse. Since GUy = C2m, one has G/GVx n L is abelian, and so GVx < G, completing the proof of the Claim.
By Lemma 3.1, we may let Qm+1 = K2 □ • • • □ K2 be a connected normal bi-Cayley
v--'
m+1 times
graph over GUy = C". By Claim, we may view r1 as a normal Cayley graph over GVx. Since r1 is coprime to K2, by Theorem 2.4, r1 □ Qm+1 is a connected normal bi-Cayley graph over Gyx x GUy = G.	□
Lemma 4.4. Let G be a group in Class E of Theorem 4.2. Then G has a normal bi-Cayley graph.
Proof. By Lemma 3.1, each of the groups in Class E (1) has a connected normal bi-Cayley graph.
Let G = D2n = (a, b | an = b2 = 1,b-1ab = a-1) with n > 3. Let r = Cay(G, {ab, b}). Then r is a cycle of length 2n, and so r is coprime to K2. By Theorem 2.4, r □ K2 is a connected normal bi-Cayley graph over G. Thus, each of the groups in Class E (2) has a connected normal bi-Cayley graph.
Let G = Alt(4) and let r = Cay(G, {(1 2 3), (1 3 2), (1 2 4), (14 2)}). By Magma [6], we have r □ K2 is a connected normal bi-Cayley graph over Alt(4).
Let G = (a, b, c | a2 = b2 = c2 = 1, abc = bca = cab) be the group in Class E (4). Let r = Cay(G, {a, b, c}). By Magma [6], r is a connected trivalent normal Cayley graph over G and r has girth 6. Hence, r is coprime to K2. By Theorem 2.4, r □ K2 is a connected normal bi-Cayley graph over G.
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Let G = (a,b | a8 = b2 = 1,bab = a5) be the group in Class E (5). Let r = Cay(G, {a, a-1, b, a4, a4b}). By [22, Lemma 6], r is a connected normal Cayley graph over G, and by Magma, Aut(r □ K2) = Aut(r) x Z2. Thus, r □ K2 is a normal bi-Cayley graph over G.
Let G = (a,b,c | a3 = b3 = c2 = 1 ,ac = ca, (ab)2 = (cb)2 = 1) be the group in Class E (6). Let r = Cay(G, {c, ca, cb}). By Magma [6], r is a connected trivalent normal Cayley graph over G and r has girth 6. Hence, r is coprime to K2. By Lemma 2.4, r □ K2 is a connected normal bi-Cayley graph over G.
Let G = (a,b,c | a3 = b3 = c3 = 1, ac = ca, bc = cb,c = a-1b-1cb) be the group in Class E (7). Let r = Cay(G, {a, b, a-1,b-1}). By Magma [6], r is a connected trivalent normal Cayley graph over G. Since G has order 27, r is coprime to K2. By Theorem 2.4, r □ K2 is a connected normal bi-Cayley graph over G.
Finally, we consider the groups in Class E (8). By Lemma 3.2, GP(8, 3) is a normal bi-Cayley graph over Q8. For n > 3, let Cn = (a) and let r = Cay(Cn, {a, a-1}). Clearly, r is a normal Cayley graph over Cn. Since GP(8,3) is of girth 6, GP(8, 3) is coprime to r. By Theorem 2.4, GP(8,3) □ r is a connected normal bi-Cayley graph over Q8 x Cn. Thus each of the groups in Class E (8) has a connected normal bi-Cayley graph.	□
Lemma 4.5. Let G be a group in Class C of Theorem 4.2. Then G has a normal bi-Cayley graph.
Proof. Since G is abelian, G has an automorphism a such that a maps every element of G to its inverse. Set H = G x (a). If H has a GRR r, then r is also a normal bi-Cayley graph over G. Suppose that H has no GRR. Then by Theorem 4.2 we have H is one of the groups in Class E (2) and (6). By Lemma 4.4, G has a normal bi-Cayley graph	□
Proof of Theorem 1.4. Let G be a finite group. If G has a GRR, then by Lemma 4.1, G has a connected normal bi-Cayley graph. If G does not have a GRR, then the theorem follows from Lemmas 4.3, 4.4, 4.5 and 3.2.	□
5 Final remarks
This paper would not be complete without mentioning some related work, namely on some special families of bi-Cayley graphs such as bi-circulants, bi-abelians etc. Numerous papers on the topic have been published (see, for instance, [1, 2, 7, 13, 14, 15, 16, 17, 18, 19, 20, 21]). In view of these, the following problem arises naturally.
Problem 5.1. For a given finite group H, classify or characterize bi-Cayley graphs with specific symmetry properties over H.
Let H be a finite group. We say that a bi-Cayley graph r of regular valency over H is a bi-graphical regular representation (or bi-GRR for short) if Aut(r) = BR(H). Motivated by the classification of finite groups having no GRR (see Theorem 4.2), we would like to pose the following problem.
Problem 5.2. Determine finite groups which have no bi-GRR.
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