The Mathieu Groups

1951 ◽  
Vol 3 ◽  
pp. 164-174 ◽  
Author(s):  
R. G. Stanton
Keyword(s):  
Simple Groups ◽  
Permutation Groups ◽  
Alternating Groups ◽  
Mathieu Groups ◽  
In Virtue Of

An enumeration of known simple groups has been given by Dickson [17]; to this list, he made certain additions in later papers [15], [16]. However, with but five exceptions, all known simple groups fall into infinite families; these five unusual simple groups were discovered by Mathieu [21], [22] and, after occasioning some discussion [20], [23], [27], were relegated to the position, which they still hold, of freakish groups without known relatives. Further interest is attached to these Mathieu groups in virtue of their providing the only known examples (other than the trivial examples of the symmetric and alternating groups) of quadruply and quintuply transitive permutation groups.

scholarly journals Simple groups, permutation groups, and probability

1999 ◽  
Vol 12 (2) ◽  
pp. 497-520 ◽  
Author(s):  
Martin W. Liebeck ◽  
Aner Shalev
Keyword(s):  
Simple Groups ◽  
Permutation Groups

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 A CONJECTURE ON THE PERMUTATION CHARACTERS OF FINITE PRIMITIVE GROUPS

2019 ◽  
Vol 102 (1) ◽  
pp. 77-90
Author(s):  
PABLO SPIGA
Keyword(s):  
Fixed Point ◽  
Finite Group ◽  
Simple Group ◽  
Simple Groups ◽  
Alternating Group ◽  
Permutation Groups ◽  
Sporadic Simple Group ◽  
Almost Simple Groups ◽  
A Finite Group ◽  
Groups Of Lie Type

Let $G$ be a finite group with two primitive permutation representations on the sets $\unicode[STIX]{x1D6FA}_{1}$ and $\unicode[STIX]{x1D6FA}_{2}$ and let $\unicode[STIX]{x1D70B}_{1}$ and $\unicode[STIX]{x1D70B}_{2}$ be the corresponding permutation characters. We consider the case in which the set of fixed-point-free elements of $G$ on $\unicode[STIX]{x1D6FA}_{1}$ coincides with the set of fixed-point-free elements of $G$ on $\unicode[STIX]{x1D6FA}_{2}$, that is, for every $g\in G$, $\unicode[STIX]{x1D70B}_{1}(g)=0$ if and only if $\unicode[STIX]{x1D70B}_{2}(g)=0$. We have conjectured in Spiga [‘Permutation characters and fixed-point-free elements in permutation groups’, J. Algebra299(1) (2006), 1–7] that under this hypothesis either $\unicode[STIX]{x1D70B}_{1}=\unicode[STIX]{x1D70B}_{2}$ or one of $\unicode[STIX]{x1D70B}_{1}-\unicode[STIX]{x1D70B}_{2}$ and $\unicode[STIX]{x1D70B}_{2}-\unicode[STIX]{x1D70B}_{1}$ is a genuine character. In this paper we give evidence towards the veracity of this conjecture when the socle of $G$ is a sporadic simple group or an alternating group. In particular, the conjecture is reduced to the case of almost simple groups of Lie type.

scholarly journals Groups with the same orders of Sylow normalizers as the Mathieu groups

2005 ◽  
Vol 2005 (9) ◽  
pp. 1449-1453 ◽  
Author(s):  
Behrooz Khosravi ◽  
Behnam Khosravi
Keyword(s):  
Finite Group ◽  
Prime Divisor ◽  
Simple Groups ◽  
Sporadic Groups ◽  
Mathieu Group ◽  
Sporadic Simple Groups ◽  
Mathieu Groups ◽  
A Finite Group

There exist many characterizations for the sporadic simple groups. In this paper we give two new characterizations for the Mathieu sporadic groups. LetMbe a Mathieu group and letpbe the greatest prime divisor of|M|. In this paper, we prove thatMis uniquely determined by|M|and|NM(P)|, whereP∈Sylp(M). Also we prove that ifGis a finite group, thenG≅Mif and only if for every primeq,|NM(Q)|=|NG(Q′)|, whereQ∈Sylq(M)andQ′∈Sylq(G).

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.

Finite classical groups and multiplication groups of loops

1995 ◽  
Vol 117 (3) ◽  
pp. 425-429 ◽  
Author(s):  
Ari Vesanen
Keyword(s):  
Permutation Group ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Classical Groups ◽  
Alternating Groups ◽  
Multiplication Group ◽  
Finite Classical Groups ◽  
Left And Right

Let Q be a loop; then the left and right translations La(x) = ax and Ra(x) = xa are permutations of Q. The permutation group M(Q) = 〈La, Ra | a ε Q〉 is called the multiplication group of Q; it is well known that the structure of M(Q) reflects strongly the structure of Q (cf. [1] and [8], for example). It is thus an interesting question, which groups can be represented as multiplication groups of loops. In particular, it seems important to classify the finite simple groups that are multiplication groups of loops. In [3] it was proved that the alternating groups An are multiplication groups of loops, whenever n ≥ 6; in this paper we consider the finite classical groups and prove the following theorems

scholarly journals Simple groups of dynamical origin

2017 ◽  
Vol 39 (3) ◽  
pp. 707-732 ◽  
Author(s):  
V. NEKRASHEVYCH
Keyword(s):  
Normal Subgroup ◽  
Group Action ◽  
Simple Groups ◽  
Full Group ◽  
Normal Subgroups ◽  
Finitely Generated ◽  
Alternating Groups ◽  
Étale Groupoid ◽  
Dynamical Origin

We associate with every étale groupoid $\mathfrak{G}$ two normal subgroups $\mathsf{S}(\mathfrak{G})$ and $\mathsf{A}(\mathfrak{G})$ of the topological full group of $\mathfrak{G}$, which are analogs of the symmetric and alternating groups. We prove that if $\mathfrak{G}$ is a minimal groupoid of germs (e.g., of a group action), then $\mathsf{A}(\mathfrak{G})$ is simple and is contained in every non-trivial normal subgroup of the full group. We show that if $\mathfrak{G}$ is expansive (e.g., is the groupoid of germs of an expansive action of a group), then $\mathsf{A}(\mathfrak{G})$ is finitely generated. We also show that $\mathsf{S}(\mathfrak{G})/\mathsf{A}(\mathfrak{G})$ is a quotient of $H_{0}(\mathfrak{G},\mathbb{Z}/2\mathbb{Z})$.

scholarly journals Invariance groups of finite functions and orbit equivalence of permutation groups

2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Eszter K. Horváth ◽  
Géza Makay ◽  
Reinhard Pöschel ◽  
Tamás Waldhauser
Keyword(s):  
Symmetric Group ◽  
Partial Answer ◽  
Permutation Groups ◽  
Computational Results ◽  
Orbit Equivalence ◽  
Alternating Groups ◽  
Primitive Groups ◽  
Complete Answer ◽  
The Difference ◽  
Invariance Groups

AbstractWhich subgroups of the symmetric group Sn arise as invariance groups of n-variable functions defined on a k-element domain? It appears that the higher the difference n-k, the more difficult it is to answer this question. For k ≤ n, the answer is easy: all subgroups of Sn are invariance groups. We give a complete answer in the cases k = n-1 and k = n-2, and we also give a partial answer in the general case: we describe invariance groups when n is much larger than n-k. The proof utilizes Galois connections and the corresponding closure operators on Sn, which turn out to provide a generalization of orbit equivalence of permutation groups. We also present some computational results, which show that all primitive groups except for the alternating groups arise as invariance groups of functions defined on a three-element domain.

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