The projective characters of the Mathieu group M12 and of its automorphism group

1980 ◽  
Vol 87 (3) ◽  
pp. 401-412 ◽  
Author(s):  
J. F. Humphreys
Keyword(s):  
Automorphism Group ◽  
Cohomology Class ◽  
Schur Multiplier ◽  
Mathieu Group ◽  
Projective Representations

In (1), Burgoyne and Fong have shown that the Schur multiplier of the Mathieu group M12 is of order 2. It is shown in Theorem 2·4 that the Schur multiplier of Aut M12, the automorphism group of M12, is also of order 2. It is therefore possible to choose a complex 2-cocycle α of Aut M12, taking only the values 1 and − 1, such that the cohomology class of α is of order 2 and the cohomology class of the restriction of α to M12 is of order 2. The characters of the irreducible α-projective representations of Aut M12 are calculated in § 2.

scholarly journals On the degrees of projective representations

1988 ◽  
Vol 30 (2) ◽  
pp. 133-135 ◽  
Author(s):  
R. J. Higgs
Keyword(s):  
Cohomology Class ◽  
Conjugacy Classes ◽  
Complex Numbers ◽  
Schur Multiplier ◽  
Projective Representations

All representations and characters studied in this paper are taken over the complex numbers, and all groups considered are finite. For basic definitions concerning projective representations see [1].If G is a group and or is a cocycle of G we denote by Proj(G, α) = {ξ1, …, ξt} the set of irreducible projective characters of G with cocycle α, where of course t is the number of α-regular conjugacy classes of G; ξ1, (1) is called the degree of ξ1. Also as normal, M(G) will denote the Schur multiplier of G, [α] the cohomology classof α, and [1] the cohomology class of the trivial cocycle of G.

scholarly journals The theta functions of sublattices of the Leech lattice

1986 ◽  
Vol 101 ◽  
pp. 151-179 ◽  
Author(s):  
Takeshi Kondo ◽  
Takashi Tasaka
Keyword(s):  
Automorphism Group ◽  
Euclidean Space ◽  
Theta Functions ◽  
Orthogonal Basis ◽  
Permutation Representation ◽  
Dimensional Euclidean Space ◽  
Unimodular Lattice ◽  
Mathieu Group ◽  
Leech Lattice ◽  
Even Unimodular Lattice

Let Λ be the Leech lattice which is an even unimodular lattice with no vectors of squared length 2 in 24-dimensional Euclidean space R24. Then the Mathieu Group M24 is a subgroup of the automorphism group .0 of Λ and the action on Λ of M24 induces a natural permutation representation of M24 on an orthogonal basis For , let Λm be the sublattice of vectors invariant under m:

Finite Groups Which are Automorphism Groups of Infinite Groups Only

1985 ◽  
Vol 28 (1) ◽  
pp. 84-90
Author(s):  
Jay Zimmerman
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Finite Field ◽  
Finite Groups ◽  
Finite Simple Group ◽  
Automorphism Groups ◽  
Outer Automorphism ◽  
Schur Multiplier ◽  
Infinite Groups ◽  
Infinite Set

AbstractThe object of this paper is to exhibit an infinite set of finite semisimple groups H, each of which is the automorphism group of some infinite group, but of no finite group. We begin the construction by choosing a finite simple group S whose outer automorphism group and Schur multiplier possess certain specified properties. The group H is a certain subgroup of Aut S which contains S. For example, most of the PSL's over a non-prime finite field are candidates for S, and in this case, H is generated by all of the inner, diagonal and graph automorphisms of S.

scholarly journals A pair of homotopy-theoretic version of TQFT’s induced by a Brown functor

2021 ◽  
pp. 2150053
Author(s):  
Minkyu Kim
Keyword(s):  
Dimension Reduction ◽  
Path Integral ◽  
Cohomology Class ◽  
Hopf Algebras ◽  
Path Integrals ◽  
Homology Theory ◽  
Projective Representations ◽  
Theoretic Version

The purpose of this paper is to study some obstruction classes induced by a construction of a homotopy-theoretic version of projective TQFT (projective HTQFT for short). A projective HTQFT is given by a symmetric monoidal projective functor whose domain is the cospan category of pointed finite CW-spaces instead of a cobordism category. We construct a pair of projective HTQFT’s starting from a [Formula: see text]-valued Brown functor where [Formula: see text] is the category of bicommutative Hopf algebras over a field [Formula: see text] : the cospanical path-integral and the spanical path-integral of the Brown functor. They induce obstruction classes by an analogue of the second cohomology class associated with projective representations. In this paper, we derive some formulae of those obstruction classes. We apply the formulae to prove that the dimension reduction of the cospanical and spanical path-integrals are lifted to HTQFT’s. In another application, we reproduce the Dijkgraaf–Witten TQFT and the Turaev–Viro TQFT from an ordinary [Formula: see text]-valued homology theory.

Real projective representations of finite groups

1990 ◽  
Vol 107 (1) ◽  
pp. 27-32 ◽  
Author(s):  
P. N. Hoffmann ◽  
J. F. Humphreys
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Schur Multiplier ◽  
Representations Of Finite Groups ◽  
Projective Representations ◽  
A Finite Group

The projective representations of a finite group G over a field K are divided into sets, each parametrized by an element of the group H2(G, Kx). The latter is the Schur multiplier M(G) when K = ℂ.

scholarly journals A characterization of the Petersen-type geometry of the McLaughlin group

2000 ◽  
Vol 128 (1) ◽  
pp. 21-44 ◽  
Author(s):  
B. BAUMEISTER ◽  
A. A. IVANOV ◽  
D. V. PASECHNIK
Keyword(s):  
Automorphism Group ◽  
Universal Cover ◽  
Classification Problem ◽  
Alternating Group ◽  
Sporadic Simple Group ◽  
Mathieu Group ◽  
Transitive Automorphism Group ◽  
The Right ◽  
Simply Connected

The McLaughlin sporadic simple group McL is the flag-transitive automorphism group of a Petersen-type geometry [Gscr ] = [Gscr ](McL) with the diagramdiagram herewhere the edge in the middle indicates the geometry of vertices and edges of the Petersen graph. The elements corresponding to the nodes from the left to the right on the diagram P33 are called points, lines, triangles and planes, respectively. The residue in [Gscr ] of a point is the P3-geometry [Gscr ](Mat22) of the Mathieu group of degree 22 and the residue of a plane is the P3-geometry [Gscr ](Alt7) of the alternating group of degree 7. The geometries [Gscr ](Mat22) and [Gscr ](Alt7) possess 3-fold covers [Gscr ](3Mat22) and [Gscr ](3Alt7) which are known to be universal. In this paper we show that [Gscr ] is simply connected and construct a geometry [Gscr ]˜ which possesses a 2-covering onto [Gscr ]. The automorphism group of [Gscr ]˜ is of the form 323McL; the residues of a point and a plane are isomorphic to [Gscr ](3Mat22) and [Gscr ](3Alt7), respectively. Moreover, we reduce the classification problem of all flag-transitive Pmn-geometries with n, m [ges ] 3 to the calculation of the universal cover of [Gscr ]˜.

Some Inequalities for the Order of the Schur Multiplier of a Pair of Groups

2008 ◽  
Vol 36 (7) ◽  
pp. 2481-2486 ◽  
Author(s):  
Mohammad Reza R. Moghaddam ◽  
Ali Reza Salemkar ◽  
Taghi Karimi
Keyword(s):  
Schur Multiplier
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