scholarly journals On minimal faithful permutation representations of finite groups

1988 ◽  
Vol 38 (2) ◽  
pp. 207-220 ◽  
Author(s):  
David Easdown ◽  
Cheryl E. Praeger
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Finite Groups ◽  
Exceptional Groups ◽  
Representations Of Finite Groups ◽  
A Finite Group

The minimal (faithful) degree μ(G) of a finite group G is the least positive integer n such that G ≲ Sn. Clearly if H ≤ G then μ(H) ≤ μ(G). However if N ◃ G then it is possible for μ(G/N) to be greater than μ(G); such groups G are here called exceptional. Properties of exceptional groups are investigated and several families of exceptional groups are given. For example it is shown that the smallest exceptional groups have order 32.

MINIMAL EXCEPTIONAL -GROUPS

2018 ◽  
Vol 98 (3) ◽  
pp. 434-438
Author(s):  
ROBERT CHAMBERLAIN
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Permutation Representation ◽  
Exceptional Group ◽  
Exceptional Groups ◽  
Representations Of Finite Groups ◽  
A Finite Group

For a finite group $G$, denote by $\unicode[STIX]{x1D707}(G)$ the degree of a minimal permutation representation of $G$. We call $G$ exceptional if there is a normal subgroup $N\unlhd G$ with $\unicode[STIX]{x1D707}(G/N)>\unicode[STIX]{x1D707}(G)$. To complete the work of Easdown and Praeger [‘On minimal faithful permutation representations of finite groups’, Bull. Aust. Math. Soc.38(2) (1988), 207–220], for all primes $p\geq 3$, we describe an exceptional group of order $p^{5}$ and prove that no exceptional group of order $p^{4}$ exists.

scholarly journals Modular Lie representations of finite groups

2004 ◽  
Vol 77 (3) ◽  
pp. 401-424 ◽  
Author(s):  
R. M. Bryant
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Finite Groups ◽  
Homogeneous Component ◽  
Module Structure ◽  
Prime Characteristic ◽  
Finite Dimensional ◽  
Representations Of Finite Groups ◽  
A Finite Group

AbstractLet K be a field of prime characteristic p and let G be a finite group with a Sylow p-subgroup of order p. For any finite-dimensional K G-module V and any positive integer n, let Ln (V) denote the nth homogeneous component of the free Lie K-algebra generated by (a basis of) V. Then Ln(V) can be considered as a K G-module, called the nth Lie power of V. The main result of the paper is a formula which describes the module structure of Ln(V) up to isomorphism.

scholarly journals On minimal faithful permutation representations of finite groups

2000 ◽  
Vol 62 (2) ◽  
pp. 311-317 ◽  
Author(s):  
L. G. Kovács ◽  
Cheryl E. Praeger
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Positive Integer ◽  
Symmetric Group ◽  
Finite Groups ◽  
Abelian Normal Subgroup ◽  
Minimal Counterexample ◽  
Representations Of Finite Groups ◽  
A Finite Group

The minimal faithful permutation degree μ(G) of a finite group G is the least positive integer n such that G is isomorphic to a subgroup of the symmetric group Sn. Let N be a normal subgroup of a finite group G. We prove that μ(G/N) ≤ μ(G) if G/N has no nontrivial Abelian normal subgroup. There is an as yet unproved conjecture that the same conclusion holds if G/N is Abelian. We investigate the structure of a (hypothetical) minimal counterexample to this conjecture.

scholarly journals Groups whose Chermak–Delgado lattice is a quasi-antichain

2019 ◽  
Vol 22 (3) ◽  
pp. 529-544
Author(s):  
Lijian An
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Finite Groups ◽  
A Finite Group

Abstract A quasi-antichain is a lattice consisting of a maximum, a minimum, and the atoms of the lattice. The width of a quasi-antichain is the number of atoms. For a positive integer {w\geq 3} , a quasi-antichain of width w is denoted by {\mathcal{M}_{w}} . In [B. Brewster, P. Hauck and E. Wilcox, Quasi-antichain Chermak–Delgado lattice of finite groups, Arch. Math. 103 2014, 4, 301–311], it is proved that {\mathcal{M}_{w}} can be the Chermak–Delgado lattice of a finite group if and only if {w=1+p^{a}} for some positive integer a and some prime p. Let t be the number of abelian atoms in {\mathcal{CD}(G)} . In this paper, we completely answer the following question: which values of t are possible in quasi-antichain Chermak–Delgado lattices?

On the Frobenius spectrum of a finite group

2017 ◽  
Vol 16 (03) ◽  
pp. 1750051 ◽  
Author(s):  
Jiangtao Shi ◽  
Wei Meng ◽  
Cui Zhang
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Finite Groups ◽  
Prime Divisor ◽  
Complete Classification ◽  
Composition Factor ◽  
Frobenius Theorem ◽  
Even Order ◽  
A Finite Group

Let [Formula: see text] be a finite group and [Formula: see text] any divisor of [Formula: see text], the order of [Formula: see text]. Let [Formula: see text], Frobenius’ theorem states that [Formula: see text] for some positive integer [Formula: see text]. We call [Formula: see text] a Frobenius quotient of [Formula: see text] for [Formula: see text]. Let [Formula: see text] be the set of all Frobenius quotients of [Formula: see text], we call [Formula: see text] the Frobenius spectrum of [Formula: see text]. In this paper, we give a complete classification of finite groups [Formula: see text] with [Formula: see text] for [Formula: see text] being the smallest prime divisor of [Formula: see text]. Moreover, let [Formula: see text] be a finite group of even order, [Formula: see text] the set of all Frobenius quotients of [Formula: see text] for even divisors of [Formula: see text] and [Formula: see text] the maximum Frobenius quotient in [Formula: see text], we prove that [Formula: see text] is always solvable if [Formula: see text] or [Formula: see text] and [Formula: see text] is not a composition factor of [Formula: see text].

scholarly journals Modular Representations of Finite Groups with Unsaturated Split (B,N)-Pairs

1980 ◽  
Vol 32 (3) ◽  
pp. 714-733 ◽  
Author(s):  
N. B. Tinberg
Keyword(s):  
Finite Group ◽  
Weyl Group ◽  
Prime Number ◽  
Semidirect Product ◽  
Finite Groups ◽  
Modular Representations ◽  
Characteristic P ◽  
Representations Of Finite Groups ◽  
A Finite Group

1. Introduction.Let p be a prime number. A finite group G = (G, B, N, R, U) is called a split(B, N)-pair of characteristic p and rank n if(i) G has a (B, N)-pair (see [3, Definition 2.1, p. B-8]) where H= B ⋂ N and the Weyl group W= N/H is generated by the set R= ﹛ω 1,… , ω n) of “special generators.”(ii) H= ⋂n∈N n-1Bn(iii) There exists a p-subgroup U of G such that B = UH is a semidirect product, and H is abelian with order prime to p.A (B, N)-pair satisfying (ii) is called a saturated (B, N)-pair. We call a finite group G which satisfies (i) and (iii) an unsaturated split (B, N)- pair. (Unsaturated means “not necessarily saturated”.)

Restricting and Inducing on Inner Products of Representations of Finite Groups

1975 ◽  
Vol 27 (6) ◽  
pp. 1349-1354
Author(s):  
G. de B. Robinson
Keyword(s):  
Finite Group ◽  
Representation Theory ◽  
Finite Groups ◽  
Reciprocity Theorem ◽  
Frobenius Reciprocity ◽  
Inner Products ◽  
Starting Point ◽  
Representations Of Finite Groups ◽  
A Finite Group

Of recent years the author has been interested in developing a representation theory of the algebra of representations [5; 6] of a finite group G, and dually of its classes [7]. In this paper Frobenius’ Reciprocity Theorem provides a starting point for the introduction of the inverses R-1 and I-1 of the restricting and inducing operators R and I. The condition under which such inverse operations are available is that the classes of G do not splitin the subgroup Ĝ. When this condition is satisfied the application of these operations to inner products is of interest.

The nilpotency index of the radicals of group algebras of finite groups whose Sylow 3-subgroups are extra-special of order 27 of exponent 3

1988 ◽  
Vol 108 (1-2) ◽  
pp. 117-132
Author(s):  
Shigeo Koshitani
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Positive Integer ◽  
Simple Group ◽  
Finite Groups ◽  
Jacobson Radical ◽  
Group Algebras ◽  
Nilpotency Index ◽  
Characteristic 3 ◽  
A Finite Group

SynopsisLet J(FG) be the Jacobson radical of the group algebra FG of a finite groupG with a Sylow 3-subgroup which is extra-special of order 27 of exponent 3 over a field F of characteristic 3, and let t(G) be the least positive integer t with J(FG)t = 0. In this paper, we prove that t(G) = 9 if G has a normal subgroup H such that (|G:H|, 3) = 1 and if H is either 3-solvable, SL(3,3) or the Tits simple group 2F4(2)'.

scholarly journals Some properties of endomorphisms in residually finite groups

1977 ◽  
Vol 24 (1) ◽  
pp. 117-120 ◽  
Author(s):  
Ronald Hirshon
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Free Product ◽  
Finite Groups ◽  
Finitely Generated ◽  
Residually Finite ◽  
Residually Finite Group ◽  
Residually Finite Groups ◽  
A Finite Group ◽  
Torsion Free

AbstractIf ε is an endomorphism of a finitely generated residually finite group onto a subgroup Fε of finite index in F, then there exists a positive integer k such that ε is an isomorphism of Fεk. If K is the kernel of ε, then K is a finite group so that if F is a non trivial free product or if F is torsion free, then ε is an isomorphism on F. If ε is an endomorphism of a finitely generated resedually finite group onto a subgroup Fε (not necessatily of ginite index in F) and if the kernel of ε obeys the minimal condition for subgroups then there exists a positive integer k such that ε is an isomorphism on Fεk.

On character values in finite groups

1977 ◽  
Vol 17 (3) ◽  
pp. 451-461 ◽  
Author(s):  
Marcel Herzog ◽  
Cheryl E. Praeger
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Finite Groups ◽  
Complex Number ◽  
Irreducible Character ◽  
Nonidentity Element ◽  
A Finite Group

Let u be a nonidentity element of a finite group G and let c be a complex number. Suppose that every nonprincipal irreducible character X of G satisfies either X(l) − X(u) = c or X(u) = 0. It is shown that c is an even positive integer and all such groups with c ≤ 8 are described.

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