ISSN 2148-838Xhttp://dx.doi.org/10.13069/jacodesmath.90728

J. Algebra Comb. Discrete Appl.
3(3) • 155–158

Received: 30 October 2015
Accepted: 06 January 2016

Journal of Algebra Combinatorics Discrete Structures and Applications

The covering number of M24
Research Article

Michael Epstein, Spyros S. Magliveras

Abstract: A finite cover C of a group G is a finite collection of proper subgroups of G such that G is equal to
the union of all of the members of C. Such a cover is called minimal if it has the smallest cardinality
among all finite covers of G. The covering number of G, denoted by σ(G), is the number of subgroups
in a minimal cover of G. In this paper the covering number of the Mathieu group M24 is shown to
be 3336.

2010 MSC: 20B40, 20D05, 20E07, 20E32

Keywords: Group theory, Group coverings, Finite simple groups

1. Introduction

A finite collection C of proper subgroups of a group G is said to be a finite cover of G if
⋃

H∈C H = G.
Of course if G is cyclic then G does not admit such a cover, but any group with a finite noncyclic
homomorphic image has a finite cover. The covering number of such a group G is denoted by σ(G),
and is defined by σ(G) = min{|C| : C is a finite cover of G}. Any cover satisfying |C| = σ(G) is called
minimal.

In [3] J. H. E. Cohn proved that if G is a finite noncyclic supersolvable group then σ(G) = p + 1,
where p is the least prime such that G has more than one subgroup of index p, and conjectured that if
G is a finite noncyclic solvable group, then σ(G) = pa + 1, where pa is the order of the smallest chief
factor of G with more than one complement in G. This conjecture was proven by Tomkinson in [11], who
suggested investigating the covering numbers of simple groups. In [2], R. Bryce, V. Fedri, and L. Serena
determined the covering numbers of some linear groups. The covering numbers of the Suzuki groups were
investigated by M. S. Lucido in [9].

A. Maróti considers alternating and symmetric groups in [10], wherein it is shown that σ(Sn) = 2n−1
if n is odd and not equal to 9, that σ(Sn) ≤ 2n−2 if n is even, and that if n is not equal to 7 or 9 then
σ(An) ≥ 2n−2 with equality if and only if n ≡ 2 (mod 4). Further results on the covering numbers of
small alternating and symmetric groups can be found in [3, 5, 7, 8].

Michael Epstein (Corresponding Author), Spyros S. Magliveras; Department of Mathematical Sciences, Florida
Atlantic University, Boca Raton, FL 33431 (email: mepstein2012@fau.edu, spyros@fau.edu).

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M. Epstein, S. Magliveras / J. Algebra Comb. Discrete Appl. 3(3) (2016) 155–158

In [6], P. E. Holmes determined the covering numbers of the Mathieu groups M11, M22, and M23,
as well as the Lyons group and the O’Nan group, and gave upper and lower bounds for the covering
numbers of the Janko group J1 and the McLaughlin group. The covering number of M12 was determined
by L. C. Kappe, D. Nikolova-Popova, and E. Swartz in [8].

The aim of this paper is to show that σ(M24) = 3336.

2. Preliminaries

Throughout we use standard terminology and notation from group theory. We will write N ·H and
N\H to denote a split extension of N by H and a general extension of N by H respectively. If π is an
element of a permutation group and the disjoint cycle decomposition of π has ki cycles of length mi,
1 ≤ i ≤ r, with m1 > m2 > ... > mr, we will write the cycle type of π as mk11 m

k2
2 ...m

kr
r .

Let G be a group and x ∈ G. If 〈x〉 is maximal among cyclic subgroups of G then we call x a principal
element and 〈x〉 a principal subgroup of G. It is easy to see that a collection C of proper subgroups of G
is a cover if and only if every principal subgroup is contained in a member of C.

If G is a finite noncyclic group and C is a finite cover of G, then by replacing each subgroup H ∈C
with a maximal subgroup M of G such that H ≤ M, we can obtain a cover C′ of G consisting of maximal
subgroups with |C′| ≤ |C|. So, for the purpose of determining the covering number of such a group it
suffices to consider covers consisting solely of maximal subgroups.

3. The Mathieu group M24

In light of the discussion in section 2, we begin with the maximal subgroups and the principal
elements of M24. As seen in [4], there are 9 conjugacy classes of maximal subgroups of M24, which we
denote by Mi, 1 ≤ i ≤ 9 ordered such that |M1| ≤ |M2| ≤ ... ≤ |M9|. The sizes of these conjugacy
classes of maximal subgroups are given by (|M1|, ..., |M9|) = (24,276,759,1288,1771,2024,3795,40320,
1457280). If Hi ∈ Mi for i = 1, ...,9 then the isomorphism types of the Hi are as follows: H1 ∼= M23,
H2 ∼= M22 ·Z2, H3 ∼= Z42 ·A8, H4 ∼= M12 ·Z2, H5 ∼= Z62 · (Z3\S6), H6 ∼= L3(4) ·S3, H7 ∼= Z62 · (L3(2)×S3),
H8 ∼= L2(23), and H9 ∼= L2(7). Let X = {j ∈ Z | 1 ≤ j ≤ 24}, and for a positive integer k let

(
X
k

)
denote

the set of all subsets of X with cardinality k. We note that H1, H2, and H6 are stabilizers in the actions
of M24 on X,

(
X
2

)
, and

(
X
3

)
respectively.

The principal elements of M24 (represented on 24 points) have cycle types 82412112, 10222, 11212,
121614121, 122, 141712111, 151513111, 21131, and 23111. We will denote the sets of principal elements
with these cycle types by T1, . . . ,T9 respectively. We remark that T6, T7, T8, and T9 are each the union
of two conjugacy classes of principal elements with the same cycle type, while the remaining Ti consist
of a single conjugacy class of elements. The cardinalities of these sets are given by (|T1|, . . . , |T9|) =
(15301440,12241152,22256640,20401920,20401920,34974720,32643072,23316480,21288960).

We describe the incidence between the sets T1, . . . ,T9 and the classes M1, . . . ,M9 of maximal sub-
groups with a matrix A = (ai,j) where the entry ai,j in row Ti and column Mj is the number of elements
from Ti contained in each maximal subgroup from class Mj. The entries of this matrix were computed
using the Magma algebra system [1], and are given in Table 1.

Observe that the elements from T1, T3, T6, T7, and T9 each fix a point of X and therefore are
contained within the subgroups from class M1. Each element from T8 has a single cycle of length 3 and
is therefore contained within a unique member of class M6. From table 1 we can see that the subgroups
from class M4 contain elements from each of T2, T4, and T5, and since each of these sets of principal
elements consists of a single conjugacy class, every element from T2 ∪T4 ∪T5 is contained within some
member of M4. Consequently, M1 ∪M4 ∪M6 is a cover of M24 by 24 + 1288 + 2024 = 3336 maximal
subgroups, and σ(M24) ≤ 3336.

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M. Epstein, S. Magliveras / J. Algebra Comb. Discrete Appl. 3(3) (2016) 155–158

Table 1. The incidence matrix A

Ti\Mj M1 M2 M3 M4 M5 M6 M7 M8 M9
T1 1275120 110880 20160 23760 8640 15120 4032 0 0
T2 0 88704 0 28512 6912 0 0 0 0
T3 1854720 80640 0 17280 0 0 0 2760 0
T4 0 73920 26880 31680 23040 0 5376 0 0
T5 0 0 0 15840 11520 0 5376 1012 0
T6 1457280 126720 46080 0 0 17280 9216 0 0
T7 1360128 0 43008 0 18432 16128 0 0 0
T8 0 0 0 0 0 11520 6144 0 0
T9 887040 0 0 0 0 0 0 528 0

Now suppose that C is a cover of M24 which consists of maximal subgroups. For 1 ≤ i ≤ 9, let
xi = |C∩Mi|. Since the subgroups from class M9 contain no principal elements, we may assume without
loss of generality that x9 = 0. Then since C is a cover of M24 we must have

8∑
j=1

ai,jxj ≥ |Ti|, 1 ≤ i ≤ 9. (1)

The reader can verify (by integer linear programming, for example) that if (x1, ...,x8) is a tuple of
nonnegative integers with xj ≤ |Mj| for 1 ≤ j ≤ 8 which satisfies the system of inequalities given by
(1), then

∑8
j=1 xj ≥ 3336. Thus for any such cover C we have |C| ≥ 3336, and so we conclude that

σ(M24) = 3336.

Acknowledgment: The authors would like to express their thanks to Dr. Igor Kliakhandler whose
generous support made possible a most significant Conference on Algebraic Combinatorics and Applica-
tions at Michigan Technical University in August, 2015. The authors also wish to thank Prof. Vladimir
Tonchev for his work to secure extra funding, a superbly organized conference, and the wonderful hospi-
tality.

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	Introduction
	Preliminaries
	The Mathieu group M24
	References