Abstract Definitions for the Mathieu Groups M11and M12

1959 ◽  
Vol 2 (1) ◽  
pp. 9-13 ◽  
Author(s):  
W.O.J. Moser
Keyword(s):  
Simple Groups ◽  
Finite Simple Groups ◽  
Exceptional Groups ◽  
Defining Relations ◽  
Mathieu Groups

A list of known finite simple groups has been given by Dickson [3, 4]. With but five exceptions, all of them fall into infinite families. The five exceptional groups, discovered by Mathieu [8,9], were further investigated by Jordan [7], Miller [10], de Séguier [11], Zassenhaus [13], and Witt [12]. In Witt's notation they are M11, M12, M22, M23, M24. Generators for them may be seen in the book of Carmichael [1, pp. 151, 263, 288]; but only for the smallest of them, M11 of order 7920, has a set of defining relations been given.

Generators for Simple Groups

1962 ◽  
Vol 14 ◽  
pp. 277-283 ◽  
Author(s):  
Robert Steinberg
Keyword(s):  
Arbitrary Element ◽  
Unitary Groups ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Lie Theory ◽  
Unimodular Group ◽  
Exceptional Groups ◽  
Mathieu Groups ◽  
Exceptional Lie Groups ◽  
Group 1

The list of known finite simple groups other than the cyclic, alternating, and Mathieu groups consists of the classical groups which are (projective) unimodular, orthogonal, symplectic, and unitary groups, the exceptional groups which are the direct analogues of the exceptional Lie groups, and certain twisted types which are constructed with the aid of Lie theory (see §§3 and 4 below). In this article, it is proved that each of these groups is generated by two of its elements. It is possible that one of the generators can be chosen of order 2, as is the case for the projective unimodular group (1), or even that one of the generators can be chosen as an arbitrary element other than the identity, as is the case for the alternating groups. Either of these results, if true, would quite likely require methods much more detailed than those used here.

scholarly journals Finite simple exceptional groups of Lie type in which all subgroups of odd index are pronormal

2020 ◽  
Vol 23 (6) ◽  
pp. 999-1016
Author(s):  
Anatoly S. Kondrat’ev ◽  
Natalia V. Maslova ◽  
Danila O. Revin
Keyword(s):  
Simple Groups ◽  
Finite Simple Groups ◽  
Exceptional Groups ◽  
Odd Index ◽  
Groups Of Lie Type

AbstractA subgroup H of a group G is said to be pronormal in G if H and {H^{g}} are conjugate in {\langle H,H^{g}\rangle} for every {g\in G}. In this paper, we determine the finite simple groups of type {E_{6}(q)} and {{}^{2}E_{6}(q)} in which all the subgroups of odd index are pronormal. Thus, we complete a classification of finite simple exceptional groups of Lie type in which all the subgroups of odd index are pronormal.

scholarly journals On the Number of p-Regular Elements in Finite Simple Groups

2009 ◽  
Vol 12 ◽  
pp. 82-119 ◽  
Author(s):  
László Babai ◽  
Péter P. Pálfy ◽  
Jan Saxl
Keyword(s):  
Simple Group ◽  
Finite Subgroup ◽  
Simple Groups ◽  
Alternating Group ◽  
Arbitrary Field ◽  
Finite Simple Groups ◽  
Regular Elements ◽  
Exceptional Groups ◽  
Monte Carlo Algorithms ◽  
A Finite Group

AbstractA p-regular element in a finite group is an element of order not divisible by the prime number p. We show that for every prime p and every finite simple group S, a fair proportion of elements of S is p-regular. In particular, we show that the proportion of p-regular elements in a finite classical simple group (not necessarily of characteristic p) is greater than 1/(2n), where n – 1 is the dimension of the projective space on which S acts naturally. Furthermore, in an exceptional group of Lie type this proportion is greater than 1/15. For the alternating group An, this proportion is at least 26/(27√n), and for sporadic simple groups, at least 2/29.We also show that for an arbitrary field F, if the simple group S is a quotient of a finite subgroup of GLn(F) then for any prime p, the proportion of p-regular elements in S is at least min{1/31, 1/(2n)}.Along the way we obtain estimates for the proportion of elements of certain primitive prime divisor orders in exceptional groups, complementing work by Niemeyer and Praeger (1998).Our result shows that in finite simple groups, p-regular elements can be found efficiently by random sampling. This is a key ingredient to recent polynomial-time Monte Carlo algorithms for matrix groups.Finally we complement our lower bound results with the following upper bound: for all n ≥ 2 there exist infinitely many prime powers q such that the proportion of elements of odd order in PSL(n,q) is less than 3/√n.

scholarly journals The Schur Multipliers of the Mathieu Groups

1966 ◽  
Vol 27 (2) ◽  
pp. 733-745 ◽  
Author(s):  
N. Burgoyne ◽  
P. Fong
Keyword(s):  
Simple Groups ◽  
Finite Simple Groups ◽  
Schur Multiplier ◽  
Lie Group Theory ◽  
Schur Multipliers ◽  
Complex Roots ◽  
Connected Covering ◽  
Group A ◽  
Mathieu Groups ◽  
Simply Connected

The Mathieu groups are the finite simple groups M11, M12, M22, M23, M24 given originally as permutation groups on respectively 11, 12, 22, 23, 24 symbols. Their definition can best be found in the work of Witt [1]. Using a concept from Lie group theory we can describe the Schur multiplier of a group as the center of a “simply-connected” covering of that group. A precise definition will be given later. We also mention that the Schur multiplier of a group is the second cohomology group of that group acting trivially on the complex roots of unity. The purpose of this paper is to determine the Schur multipliers of the five Mathieu groups.

On the Mathieu group M23

1971 ◽  
Vol 12 (4) ◽  
pp. 385-392 ◽  
Author(s):  
N. Bryce
Keyword(s):  
Simple Group ◽  
Finite Simple Group ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Unique Determination ◽  
Mathieu Group ◽  
Mathieu Groups

Until 1965, when Janko [7] established the existence of his finite simple group J1, the five Mathieu groups were the only known examples of isolated finite simple groups. In 1951, R. G. Stanton [10] showed that M12 and M24 were determined uniquely by their order. Recent characterizations of M22 and M23 by Janko [8], M22 by D. Held [6], and M11 by W. J. Wong [12], have facilitated the unique determination of the three remaining Mathieu groups by their orders. D. Parrott [9] has so characterized M22 and M11, while this paper is an outline of the characterization of M23 in terms of its order.

Natural constructions of the Mathieu groups

1989 ◽  
Vol 106 (3) ◽  
pp. 423-429 ◽  
Author(s):  
R. T. Curtis
Keyword(s):  
Symmetric Groups ◽  
The Other ◽  
Simple Groups ◽  
Alternating Group ◽  
Transitive Permutation Group ◽  
French Mathematician ◽  
Exceptional Groups ◽  
Transitive Groups ◽  
Mathieu Groups ◽  
Alternating And Symmetric Groups

In the second half of the last century the French mathematician Emil Mathieu discovered two quintuply transitive permutation groups, now labelled M12 and M24, acting on twelve and twenty-four letters respectively. With the classification of finite simple groups complete we now know that any other quintuply transitive permutation group, on any number of letters, must contain the corresponding alternating group. Indeed, the only quadruply transitive groups, other than the alternating and symmetric groups, are the point stabilizers in M12 and M24, which are denoted by M11 and M23 respectively. To put it another way, the study of multiply (≥ 4-fold) transitive groups now means the study of the symmetric groups and the Mathieu groups. Apart from their beauty and interest in their own right the Mathieu groups are involved in many of the other sporadic simple groups: see ([2], p. 238). Thus a detailed understanding of the other exceptional groups necessitates an intimate knowledge of M12 and M24.

scholarly journals Presentations of some finite simple groups

1988 ◽  
Vol 45 (2) ◽  
pp. 143-168 ◽  
Author(s):  
Dragomir Ž. Đoković
Keyword(s):  
Simple Groups ◽  
Finite Simple Groups ◽  
Mathieu Groups

AbstractWe give new presentations of the five Mathieu groups, the simple groups J1, J2, HS, McL, Co3, and some other simple and related groups. All generators in these presentations are involutions. Our presentations are simpler than the known presentations of this type for the groups mentioned above.

scholarly journals A diameter bound for finite simple groups of large rank

2017 ◽  
Vol 95 (2) ◽  
pp. 455-474 ◽  
Author(s):  
Arindam Biswas ◽  
Yilong Yang
Keyword(s):  
Simple Groups ◽  
Finite Simple Groups ◽  
Large Rank

scholarly journals New Beauville surfaces and finite simple groups

2013 ◽  
Vol 142 (3-4) ◽  
pp. 391-408 ◽  
Author(s):  
Shelly Garion ◽  
Matteo Penegini
Keyword(s):  
Simple Groups ◽  
Finite Simple Groups ◽  
Beauville Surfaces
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