ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 23 (2023) #P2.03
https://doi.org/10.26493/1855-3974.2415.fd1
(Also available at http://amc-journal.eu)
Classification of minimal Frobenius hypermaps*
Kai Yuan , Yan Wang †
School of Mathematics and Information Science, Yan Tai University, Yan Tai, P.R.C.
Received 23 August 2020, accepted 24 May 2022, published online 11 November 2022
Abstract
In this paper, we give a classification of orientably regular hypermaps with an automor-
phism group that is a minimal Frobenius group. A Frobenius group G is called minimal if
it has no nontrivial normal subgroup N such that G{N is a Frobenius group. An orientably
regular hypermap H is called a Frobenius hypermap if AutpHq acting on the hyperfaces
is a Frobenius group. A minimal Frobenius hypermap is a Frobenius hypermap whose
automorphism group is a minimal Frobenius group with cyclic point stabilizers. Every
Frobenius hypermap covers a minimal Frobenius hypermap. The main theorem of this
paper generalizes the main result of Breda D’Azevedo and Fernandes in 2011.
Keywords: Frobenius hypermap, Frobenius group.
Math. Subj. Class. (2020): 57M15, 05C25, 20F05
1 Introduction
Let S be a compact and connected orientable surface. A topological hypermap H on S is a
triple pS;V ;Eq, where V and E denote closed subsets of S with the following properties:
(1) B “ V X E is a finite set. Its elements are called the brins of H;
(2) V Y E is connected;
(3) the components of V (called the hypervertices) and of E (called the hyperedges), are
homeomorphic to closed discs;
(4) the components of the complement SzpV YEq are homeomorphic to open discs, and
they are called the hyperfaces of H.
*The Authors thank the referees for their helpful comments.
†Corresponding author. Supported by NSFC (No. 12101535) and NSFS (No. ZR2020MA044).
E-mail addresses: pktide@163.com (Kai Yuan), wang´yan@pku.org.cn (Yan Wang)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
2 Ars Math. Contemp. 23 (2023) #P2.03
The following Figure 1 shows a topological hypermap on torus with 9 brins, 3 hy-
pervertices (black components), 3 hyperedges (grey components) and 3 hyperfaces (white
components).
Figure 1: A hypermap on torus.
An important and convenient way to visualize hypermaps was introduced by Walsh in
[13]. The Walsh representation of a hypermap as a bipartite graph embedding on S can
be described as follows. At the centre of each hypervertex place a white vertex and at the
centre of each hyperedge place a black vertex. If a hypervertex intersects a hyperedge then
we join the corresponding white vertex and black vertex by an edge. In this way we obtain
a bipartite graph. This bipartite graph is said to be the underlying graph of H. Figure 2 is
the Walsh representation of the hypermap in Figure 1.
Figure 2: The Walsh representation.
An algebraic hypermap is a quadruple H “ pG,B, ρ0, ρ1q, where G is a finite group
which is generated by two elements ρ0, ρ1 and acts transitively on a finite set B. By [3],
there is a one-to-one correspondence between topological and algebraic hypermaps. The
finite group G is the monodromy group of H, denoted by MonpHq. In the Walsh repre-
sentation, G is a permutation group acting on the set of edges, ρ0, ρ1 generate the cyclic
permutations of the edges going around the white resp. black vertices in a positive sense,
and each cycle of ρ0ρ1 bounds a hyperface in a negative direction. A permutation α of B
is called an automorphism of the hypermap H “ pG,B, ρ0, ρ1q if it is G-equivariant, i.e.
if
αpgpbqq “ gpαpbqq
K. Yuan and Y. Wang: Classification of minimal Frobenius hypermaps 3
for every b P B and g P G.
Since αρ0α´1 “ ρ0 and αρ1α´1 “ ρ1, α induces a permutation on the cycles of ρ0 and
ρ1. So, in the Walsh representation, AutpHq induces a subgroup of the automorphism group
of the underlying graph of H, and AutpHq preserves the hypervertex set and hyperedge set,
respectively. A hypermap is called regular if G acts regularly on B. In this case, AutpHq
is isomorphic to G which acts regularly on B as well.
For a regular hypermap H “ pG,B, ρ0, ρ1q, the set B can be replaced by G, so that
MonpHq and AutpHq can be viewed as the right and left regular multiplications of G,
respectively. So, H can be denoted by a triple H “ pG; ρ0, ρ1q, where G “ xρ0, ρ1y. In
this way, the hypervertices (resp. hyperedges and hyperfaces ) correspond to right cosets
of G relative to xρ0y, (resp. xρ1y and xρ0ρ1y). In [4], the hypermap H “ pG; ρ0, ρ1q is
denoted by pG; a, bq where a “ ρ1´1ρ0´1 and b “ ρ0. From now on, we denote a regular
hypermap H by the triple H “ pG; a, bq, and then the hyperfaces (resp. hypervertices and
hyperedges) correspond to left cosets of G relative to subgroups xay (resp. xby and xaby).
Let H “ pG; a, bq and H1 “ pG1; a1, b1q be two orientably regular hypermaps. If there is an
epimorphism ρ from G to G1 such that aρ “ a1 and bρ “ b1, then H is called a covering of
H1 or H covers H1. Given a group G, pG; a1, b1q – pG; a2, b2q if and only if there exists
an automorphism σ of G such that aσ1 “ a2 and bσ1 “ b2.
A (face-)primer hypermap is an orientably regular hypermap whose automorphism
group induces faithful actions on its hyperfaces, see [4]. The classification of regular hy-
permaps with given automorphism groups isomorphic to PSLp2, qq or PGLp2, qq can be
extracted from [12] by Sah. Moreover, Conder, Potočnik and Širáň extended Sah’s investi-
gation to reflexible hypermaps, on both orientable and nonorientable surfaces, and provided
explicit generating sets for projective linear groups, see [1]. In [2], Conder described all
regular hypermaps of genus 2 to 101, and all non-orientable regular hypermaps of genus 3
to 202.
The study of primer hypermaps was initiated by Breda d’Azevedo and Fernandes in
2011. In [4], the authors classified the primer hypermaps with p-hyperfaces for a prime
number p, where their automorphism groups are Frobenius groups. Thereafter, they de-
termined all regular hypermaps with p-hyperfaces, see [5]. In [7], Du and Hu classified
primer hypermaps with a product of two primes number of hyperfaces. Recently, Du and
Yuan characterized primer hypermaps with nilpotent automorphism groups and prime hy-
pervertex valency, see [8].
A Frobenius group is a transitive permutation group G on a set Ω which is not regular
on Ω , but has the property that the only element of G which fixes more than one point is
the identity element. A Frobenius group G is called minimal if it does not have a nontrivial
normal subgroup N such that G{N is a Frobenius group. A regular hypermap H is called
a Frobenius hypermap if AutpHq acting on the hyperfaces is a Frobenius group. Clearly,
H is a primer hypermap. A minimal Frobenius hypermap is a Frobenius hypermap whose
automorphism group is a minimal Frobenius group with a cyclic point stabilizer. Clearly,
every Frobenius hypermap covers a minimal Frobenius hypermap.
This paper has three sections. In the first section, a quick overview of orientably regu-
lar hypermaps is given. In Section 2, we introduce minimal Frobenius groups. In the last
section, we give a classification of orientably regular minimal Frobenius hypermaps. Fur-
thermore, the main theorem of this paper generalizes the main result of Breda D’Azevedo
and Fernandes, see [4].
4 Ars Math. Contemp. 23 (2023) #P2.03
2 Minimal Frobenius groups
We refer the readers to [10] for standard notation and results in group theory. Set pr, sq
to denote the greatest common divisor of two positive integers r and s. We denote the
orders of an element x and of a subgroup H of G as |x| and |H|, respectively. A semidirect
product of a group N by a group H is denoted by N : H . Let Zm “ t0, 1, ¨ ¨ ¨ ,m ´ 1u
and Z˚m “ tk
ˇ
ˇ k P Zm and pk,mq “ 1u.
Let G be a Frobenius group on Ω. A subgroup K of G is called the Frobenius kernel
if K acts regularly on Ω. Each point stabilizer is called a Frobenius complement of K in
G. In the following, we give some interesting results about Frobenius groups and primitive
groups.
Proposition 2.1 ([6, P86]). Let G be a Frobenius group on Ω and α P Ω, K be the
Frobenius kernel, and H be a Frobenius complement. Then:
(i) K is a normal and regular subgroup of G.
(ii) For each odd prime number p, the Sylow p-subgroups of H are cyclic, and the Sylow
2-subgroups are either cyclic or quaternion groups. If G is not solvable, then it has
exactly one nonabelian composition factor, namely A5.
(iii) K is a nilpotent group.
Proposition 2.2 ([6, Corollary 1.5A.]). Let G be a group acting transitively on a set Ω
with at least two points. Then G is primitive if and only if each point stabilizer Gα is a
maximal subgroup of G.
Lemma 2.3. Assume G ď SympΩq has a regular normal subgroup R, where Ω has at least
two points. Then G is primitive if and only if no nontrivial subgroup of R is normalized by
Gα, for each α.
Proof. By Proposition 2.2, G is primitive if and only if Gα is a maximal subgroup of G.
Because R is a regular normal subgroup of G, G “ GαR and Gα X R “ t1u.
We claim that Gα is maximal if and only if no nontrivial subgroup of R is normalized
by Gα. Suppose Gα is not maximal, then there exists a proper subgroup K of G such that
Gα ă K. It follows that K “ K XG “ K XGαR “ GαpK XRq. In this case, K XR is
a proper subgroup of R which is normalized by Gα. Conversely, suppose that there exists
a proper subgroup H , normalized by Gα, of R. Thus GαH is a proper subgroup of G and
so Gα is not maximal. l
Corollary 2.4 follows directly from Lemma 2.3.
Corollary 2.4. Assume G ď SympΩq has a regular normal subgroup R, where Ω has at
least two points. If R is abelian, then G is primitive if and only if no nontrivial normal
subgroup of G is contained in R.
Lemma 2.5. Let K be the Frobenius kernel of a Frobenius group G which acts on a set Ω.
If N is a normal subgroup of G, then either N ď K or K ă N .
Proof. Assume that N is not a subgroup of K. Set α P Ω. Since N is a normal subgroup
of G, we have N “ p
Ť
gPK
Ngαq Y pN X Kq and so N is a subgroup of NαK. Let |Nα| “
K. Yuan and Y. Wang: Classification of minimal Frobenius hypermaps 5
m, |K| “ n and |N XK| “ t. Then, |N | “ npm´1q ` t. Since N ď NαK and Nα ď N ,
we get N “ N X NαK “ NαpN X Kq. So, |N | “ mt which implies npm ´ 1q ` t “
mt. Note that m ą 1, then n “ t. Therefore, N X K “ K and K is a proper subgroup of
N . l
Proposition 2.6 ([11, Lemma 2.3]). Let K be the Frobenius kernel of a Frobenius group
G. If N is a normal subgroup of G and N ă K, then G{N is a Frobenius group.
Proposition 2.7 ([11, Corollary 2.6]). Let G “ KH be a Frobenius group, where K is the
Frobenius kernel and H is a Frobenius complement. For each h P H,h ‰ 1, and for each
k P K, the orders of h, kh and hk are equal, that is |h| “ |kh| “ |hk|.
Based on Lemma 2.5 and Proposition 2.6, we give the following definition of minimal
Frobenius groups.
Definition 2.8. A Frobenius group G is called minimal if it does not have a nontrivial
normal subgroup N such that G{N is a Frobenius group.
Lemma 2.9. If G is a minimal Frobenius group acting on a set Ω with the Frobenius kernel
K, then K is an elementary abelian p-group and G is primitive.
Proof. If G is minimal, then by Proposition 2.6 no nontrivial normal subgroup of G exists
in K. Note that K is a nilpotent group. Let P be a Sylow p-group of K, ΦpP q be the
Frattini subgroup of P and L be the p1-Hall group of K. Both ΦpP q and L are characteristic
subgroups of K. So, L “ ΦpP q “ 1 which implies that K is an elementary abelian p-
group.
Because no nontrivial normal subgroup of G is contained in K and K is abelian, it
follows that G is primitive by Corollary 2.4. l
Lemma 2.10. If G is a primitive group acting on a set Ω with non-trivial abelian point sta-
bilizers, then G is a Frobenius group and its Frobenius kernel K is an elementary abelian
p-group.
Proof. It suffices to show that for any two distinct points α, β P Ω, Gα XGβ “ 1. Let J “
Gα X Gβ . Since G is primitive, G “ xGα, Gβy. Note that Gα and Gβ are abelian, so J is
a normal subgroup of G. Because αJ “ tαu, for any g P G, we have αgJ “ αJg “ tαgu.
That is to say J fixes every point of Ω, so J “ 1 and G is a Frobenius group. Furthermore,
as point stabilizers are maximal, the Frobenius kernel K must be an elementary abelian
p-group . l
Corollary 2.11 follows from Lemma 2.9 and 2.10 directly.
Corollary 2.11. Let G be a permutation group with cyclic point stabilizers. Then, G is a
minimal Frobenius group if and only if G is a primitive group.
For a prime number p and an integer n, an integer m pm ą 1q is called a primitive
divisor of pn ´ 1 if m divides pn ´ 1, but it does not divide ps ´ 1 for any s ă n.
The following Proposition 2.12 can be obtained from some results in
[10, Kapitel II: 3.10, 3.11, 7.3].
Proposition 2.12. For a prime number p and a positive integer n, set G “ GLpn, pq.
6 Ars Math. Contemp. 23 (2023) #P2.03
(i) The group G contains a cyclic Singer-Zyklus group S “ xxy of order pn ´ 1, and
CGpSq “ S. Moreover, NGpSq “ S : xyy “ xx, y
ˇ
ˇ xp
n´1 “ yn “ 1, xy “ xpy,
and |NGpSq| “ nppn ´ 1q. Take an element g P S, if |g| is a primitive divisor of
pn ´ 1, then NGpxgyq “ NGpSq, CGpxgyq “ S and xgy is an irreducible subgroup.
(ii) Let L be a cyclic irreducible subgroup of G. Then L is conjugate to a subgroup of
S, and |L| is a primitive divisor of pn ´ 1.
The following lemma generalizes Lemma 3.3 in [9]. The proof is similar to that of
Lemma 3.3, so we omit it.
Lemma 2.13. Let X “ T : xxy and Y “ T : xyy be two subgroups of A “ AGLpn, pq “
T : G, where G “ GLpn, pq, T is the translation subgroup, and x, y are nontrivial ele-
ments in G. If σ is an isomorphism from X to Y mapping xxy to xyy, then, there exists an
element u P G such that σ “ Ipuq|X , where Ipuq is the inner automorphism of A induced
by u. In particular, u P NGpxxyq if xxy “ xyy.
3 Classification of minimal Frobenius hypermaps
For a prime number p, an integer n ě 1 (n ě 2 if p “ 2) and a primitive divisor m of
pn ´ 1, let S be the cyclic Singer-Zyklus group of GLpn, pq, xay be a subgroup of S with
order m and T be the translation subgroup of AGLpn, pq. Define a group M of order mpn
as
M “ T : xay ď T : S ď AGLpn, pq “ T : GLpn, pq.
By Proposition 2.12, xay is an irreducible subgroup. Hence M is a primitive group, and
consequently M is a Frobenius group by Lemma 2.10.
Let F be a minimal Frobenius group acting on a set Ω (|Ω| ą 2) with cyclic point
stabilizers, and K be its Frobenius kernel. By Lemma 2.9, K is an elementary abelian
p-group and F is a primitive group. Set |K| “ pn, and then |Ω| “ pn. Take an element
α P Ω and assume |Fα| “ k. By Proposition 2.12, k is a primitive divisor of pn ´ 1, and
GLpn, pq has only one conjugacy class of irreducible cyclic subgroups of order k. Hence
AGLpn, pq has only one conjugacy class of subgroups isomorphic to F which implies
F – M “ T : xay when k “ m. These discussions give the following Theorem 3.1.
Theorem 3.1. Let F be a minimal Frobenius group with cyclic point stabilizers of order
m. Then, F – T : xay, where T is elementary abelian of order pn for some prime number
p and an integer n ě 1, m is a primitive divisor of pn ´ 1 and |xay| “ m. Clearly,
|F | “ mpn.
Lemma 3.2. Let M “ T : xay be the group defined as in the first paragraph of this section.
If H “ pM ;R,Lq is a Frobenius hypermap, then H is isomorphic to
Hpp, n,m, i, jq “ pM ; ai, ajbq,
where 1 ‰ b P T , m is a primitive divisor of pn ´ 1, j P Zm, i P Z˚m and pi, pq “ 1. More-
over, different parameter pairs pi, jq give non-isomorphic hypermaps with pn hyperfaces,
each of valency m. Furthermore, there are mϕpmqn non-isomorphic hypermaps, where ϕ is
the Euler’s totient function.
K. Yuan and Y. Wang: Classification of minimal Frobenius hypermaps 7
Proof. Let G “ GLpn, pq and then M ď AGLpn, pq “ T : G. Since M is a Frobenius
group, M has only one conjugacy class of subgroups of order m. So we can assume R “ ai
for some i P Z˚m. Remember that S is the cyclic Singer-Zyklus group of GLpn, pq and xay
is a subgroup of S. So, M is a normal subgroup of T : S. Since S fixes a and acts
transitively on T zt1u by conjugation, we may fix L “ ajb, where j is calculated modular
m.
If there exists an automorphism σ of M such that paiqσ “ ai1 and pajbqσ “ aj1b, then
bσ “ aϵb for some ϵ P Zm. Clearly, the orders of b and aϵb are equal. While according to
Proposition 2.7, the two elements aϵb and aϵ have the same order which is coprime with
that of b if ϵ ‰ 0 modulo m. So, bσ “ b. By Lemma 2.13, there exists an element u P G
such that σ “ Ipuq|F , where u P NGpxayq. According to Proposition 2.12,
NGpxayq “ S : xyy “ xx, y
ˇ
ˇ xp
n´1 “ yn “ 1, xy “ xpy,
where S “ xxy. Because bσ “ b, it follows that u “ yt, where t is calculated modular n.
So, aσ “ ayt “ apt . As a result, we may assume pi, pq “ 1 in R “ ai. As a result, we get
mϕpmq
n non-isomorphic hypermaps pM ; a
i, ajbq, where ϕ is the Euler’s totient function.
Clearly, pM ; ai, ajbq has pn hyperfaces, each of valency m. l
By Theorem 3.1, the automorphism group of a minimal Frobenius hypermap is isomor-
phic to M “ T : xay, where |T | “ pn and |xay| “ m. Consequently, we give the following
classification theorem of minimal Frobenius hypermaps.
Theorem 3.3. H is a minimal Frobenius hypermap if and only if H is isomorphic to
Hpp, n,m, i, jq “ pM ; ai, ajbq,
where M is a group defined as in the first paragraph of this section, m is a primitive divisor
of pn ´ 1, j P Zm, i P Z˚m and pi, pq “ 1. Moreover, different parameter pairs pi, jq give
non-isomorphic hypermaps with pn hyperfaces, each of valency m. And, there are mϕpmqn
non-isomorphic minimal Frobenius hypermaps, where ϕ is the Euler’s totient function.
According to Corollary 2.11, we have the following Proposition 3.4.
Proposition 3.4. If H is a regular hypermap, then H is a minimal Frobenius hypermap if
and only if AutpHq acts primitively on the hyperfaces.
The next Proposition 3.5 follows from Lemma 2.5.
Proposition 3.5. Every Frobenius hypermap covers a minimal Frobenius hypermap.
The H-sequence of a hypermap H is a sequence r|v|, |e|, |f |;V,E, F ; |AutpHq|s, where
|v|, |e|, |f |, V, E and F stand for the hypervertex valency, hyperedge valency, hyperface
valency, number of hypervertices, number of hyperedges and number of hyperfaces of H,
respectively.
Corollary 3.6. The H-sequence of the minimal Frobenius hypermap Hpp, n,m, i, jq “
pM ; ai, ajbq is
(i) rp,m,m;mpn´1, pn, pn;mpns for j “ 0;
(ii) rm, p,m; pn,mpn´1, pn;mpns for j “ m ´ i;
8 Ars Math. Contemp. 23 (2023) #P2.03
(iii) r mpm,jq ,
m
pm,i`jq ,m; pm, jqp
n, pm, i ` jqpn, pn;mpns for j ‰ 0 and j ‰ m ´ i.
Proof. The sequence is determined by the first three entries, namely |ajb|, |ai`jb| and |ai|.
These entries can be easily calculated according to Proposition 2.7. l
ORCID iDs
Kai Yuan https://orcid.org/0000-0003-1858-3083
Yan Wang https://orcid.org/0000-0002-0148-2932
References
[1] M. Conder, P. Potočnik and J. Širáň, Regular hypermaps over projective linear groups, J. Aust.
Math. Soc. 85 (2008), 155–175, doi:10.1017/s1446788708000827, https://doi.org/
10.1017/s1446788708000827.
[2] M. D. E. Conder, Regular maps and hypermaps of Euler characteristic ´1 to ´200, J. Comb.
Theory, Ser. B 99 (2009), 455–459, doi:10.1016/j.jctb.2008.09.003, https://doi.org/
10.1016/j.jctb.2008.09.003.
[3] D. Corn and D. Singerman, Regular hypermaps, Eur. J. Comb. 9 (1988), 337–351, doi:10.1016/
s0195-6698(88)80064-7, https://doi.org/10.1016/s0195-6698(88)80064-7.
[4] A. B. D’Azevedo and M. E. Fernandes, Classification of primer hypermaps with a prime
number of hyperfaces, Eur. J. Comb. 32 (2011), 233–242, doi:10.1016/j.ejc.2010.09.003,
https://doi.org/10.1016/j.ejc.2010.09.003.
[5] A. B. D’Azevedo and M. E. Fernandes, Classification of the regular oriented hypermaps
with prime number of hyperfaces, Ars Math. Contemp. 10 (2016), 193–209, doi:10.26493/
1855-3974.657.77e, https://doi.org/10.26493/1855-3974.657.77e.
[6] J. Dixon and B. Mortimer, Permutation Groups, Springer, NewYork, 1996, doi:10.1007/
978-1-4612-0731-3, https://doi.org/10.1007/978-1-4612-0731-3.
[7] S. Du and X. Hu, A classification of primer hypermaps with a product of two primes number
of hyperfaces, Eur. J. Comb. 62 (2017), 245–262, doi:10.1016/j.ejc.2017.01.005, https://
doi.org/10.1016/j.ejc.2017.01.005.
[8] S. Du and K. Yuan, Nilpotent primer hypermaps with hypervertices of valency a prime, J. Al-
gebr. Comb. 52 (2020), 299–316, doi:10.1007/s10801-019-00903-9, https://doi.org/
10.1007/s10801-019-00903-9.
[9] S.-F. Du, J. H. Kwak and R. Nedela, A classification of regular embeddings of graphs of order a
product of two primes, J. Algebr. Comb. 19 (2004), 123–141, doi:10.1023/b:jaco.0000023003.
69690.18, https://doi.org/10.1023/b:jaco.0000023003.69690.18.
[10] B. Huppert, Endliche Gruppen I, Springer-Verlag, Berlin, 1967, doi:10.1007/
978-3-642-64981-3, https://doi.org/10.1007/978-3-642-64981-3.
[11] H.-P. Qu, Y. Wang and K. Yuan, Frobenius groups which are the automorphism groups of
orientably-regular maps, Ars Math. Contemp. 19 (2020), 363–374, doi:10.26493/1855-3974.
1851.b44, https://doi.org/10.26493/1855-3974.1851.b44.
[12] C. H. Sah, Groups related to compact Riemann surfaces, Acta Math. 123 (1969), 13–42, doi:
10.1007/bf02392383, https://doi.org/10.1007/bf02392383.
[13] T. R. S. Walsh, Hypermaps versus bipartite maps, J. Comb. Theory, Ser. B 18 (1975), 155–163,
doi:10.1016/0095-8956(75)90042-8, https://doi.org/10.1016/0095-8956(75)
90042-8.