Outer automorphisms centralizing every abelian subgroup of a finite group

1974 ◽  
Vol 137 (2) ◽  
pp. 93-127 ◽  
Author(s):  
Everett C. Dade
Keyword(s):  
Finite Group ◽  
Abelian Subgroup ◽  
Outer Automorphisms ◽  
A Finite Group

The largest Irreducible Character Degree of a Finite Group

1985 ◽  
Vol 37 (3) ◽  
pp. 442-451 ◽  
Author(s):  
David Gluck
Keyword(s):  
Finite Group ◽  
Irreducible Character ◽  
Abelian Subgroup ◽  
Character Degree ◽  
Character Table ◽  
Significant Information ◽  
Famous Theorem ◽  
Frobenius Reciprocity ◽  
A Finite Group ◽  
Irreducible Character Degree

Much information about a finite group is encoded in its character table. Indeed even a small portion of the character table may reveal significant information about the group. By a famous theorem of Jordan, knowing the degree of one faithful irreducible character of a finite group gives an upper bound for the index of its largest normal abelian subgroup.Here we consider b(G), the largest irreducible character degree of the group G. A simple application of Frobenius reciprocity shows that b(G) ≧ |G:A| for any abelian subgroup A of G. In light of this fact and Jordan's theorem, one might seek to bound the index of the largest abelian subgroup of G from above by a function of b(G). If is G is nilpotent, a result of Isaacs and Passman (see [7, Theorem 12.26]) shows that G has an abelian subgroup of index at most b(G)4.

FINITE GROUPS WITH MULTIPLICITY-FREE PERMUTATION CHARACTERS

2005 ◽  
Vol 04 (02) ◽  
pp. 187-194
Author(s):  
MICHITAKU FUMA ◽  
YASUSHI NINOMIYA
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Abelian Subgroup ◽  
Hecke Algebra ◽  
Endomorphism Algebra ◽  
Theoretic Method ◽  
A Finite Group ◽  
Multiplicity Free ◽  
Canonical Involution

Let G be a finite group and H a subgroup of G. The Hecke algebra ℋ(G,H) associated with G and H is defined by the endomorphism algebra End ℂ[G]((ℂH)G), where ℂH is the trivial ℂ[H]-module and (ℂH)G = ℂH⊗ℂ[H] ℂ[G]. As is well known, ℋ(G,H) is a semisimple ℂ-algebra and it is commutative if and only if (ℂH)G is multiplicity-free. In [6], by a ring theoretic method, it is shown that if the canonical involution of ℋ(G,H) is the identity then ℋ(G,H) is commutative and, if there exists an abelian subgroup A of G such that G = AH then ℋ(G,H) is commutative. In this paper, by a character theoretic method, we consider the commutativity of ℋ(G,H).

scholarly journals CONJUGATE PAIRS OF SUBFACTORS AND ENTROPY FOR AUTOMORPHISMS

2011 ◽  
Vol 22 (04) ◽  
pp. 577-592 ◽  
Author(s):  
MARIE CHODA
Keyword(s):  
Finite Group ◽  
Crossed Product ◽  
View Point ◽  
Jones Index ◽  
Outer Automorphisms ◽  
Outer Action ◽  
A Finite Group

Based on the fact that, for a subfactor N of a II1factor M, the first nontrivial Jones index is two and then M is decomposed as the crossed product of N by an outer action of ℤ2, we study pairs {N, uNu*} from the view-point of entropy for two subalgebras of M in connection with the entropy for automorphisms, where the inclusion of II1factors N ⊂ M is given with M being the crossed product of N by a finite group of outer automorphisms and u is a unitary in M.

Finite groups with each non-normal subgroup having maximal normal cores

2016 ◽  
Vol 15 (03) ◽  
pp. 1650053
Author(s):  
Heng Lv ◽  
Zhibo Shao ◽  
Wei Zhou
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Abelian Subgroup ◽  
A Finite Group

In this paper, we study a finite group [Formula: see text] such that [Formula: see text] is a prime for each non-normal subgroup [Formula: see text] of [Formula: see text]. We prove that such a group must contain a big abelian subgroup. More specifically, if such a group [Formula: see text] is not supersoluble, then there is an abelian subgroup [Formula: see text] such that [Formula: see text], and if [Formula: see text] is supersoluble, then there is an abelian subgroup [Formula: see text] such that [Formula: see text] or [Formula: see text], where [Formula: see text] and [Formula: see text] are primes.

scholarly journals Rigid automorphisms of linking systems

2021 ◽  
Vol 9 ◽  
Author(s):  
George Glauberman ◽  
Justin Lynd
Keyword(s):  
Finite Group ◽  
Sylow Subgroup ◽  
Fusion Systems ◽  
Outer Automorphisms ◽  
Inner Automorphism ◽  
Inner Automorphisms ◽  
A Finite Group

Abstract A rigid automorphism of a linking system is an automorphism that restricts to the identity on the Sylow subgroup. A rigid inner automorphism is conjugation by an element in the center of the Sylow subgroup. At odd primes, it is known that each rigid automorphism of a centric linking system is inner. We prove that the group of rigid outer automorphisms of a linking system at the prime $2$ is elementary abelian and that it splits over the subgroup of rigid inner automorphisms. In a second result, we show that if an automorphism of a finite group G restricts to the identity on the centric linking system for G, then it is of $p'$ -order modulo the group of inner automorphisms, provided G has no nontrivial normal $p'$ -subgroups. We present two applications of this last result, one to tame fusion systems.

Finite groups whose non-abelian self-centralizing subgroups are TI-subgroups or subnormal subgroups

2020 ◽  
pp. 2150040
Author(s):  
Yuqing Sun ◽  
Jiakuan Lu ◽  
Wei Meng
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Abelian Subgroup ◽  
Subnormal Subgroup ◽  
A Finite Group ◽  
Subnormal Subgroups

In this paper, we prove that if every non-abelian self-centralizing subgroup of a finite group [Formula: see text] is a TI-subgroup or a subnormal subgroup of [Formula: see text], then every non-abelian subgroup of [Formula: see text] must be subnormal in [Formula: see text].

Finite Groups in Which the Automizers of Some Abelian Subgroups are Trivial

2018 ◽  
Vol 25 (04) ◽  
pp. 701-712
Author(s):  
Pengfei Bai ◽  
Xiuyun Guo
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Abelian Subgroup ◽  
Group Of Automorphisms ◽  
Normal Abelian Subgroup ◽  
Abelian Subgroups ◽  
A Finite Group

If H is a subgroup of a finite group G, then the automizer AutG(H) of H in G is defined as the group of automorphisms of H induced by conjugation by elements of NG(H). A finite group G is called an NNC-group if for any non-normal abelian subgroup A, either [Formula: see text] or [Formula: see text]. In this paper, classifications of nilpotent NNC-groups and non-solvable NNC-groups are given. We also investigate the solvable NNC-groups and describe the structure of solvable NNC-groups.

Finite groups of matrices over division rings

1982 ◽  
Vol 92 (1) ◽  
pp. 55-64 ◽  
Author(s):  
B. Hartley ◽  
M. A. Shahabi Shojaei
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Abelian Subgroup ◽  
Classical Theorem ◽  
Division Rings ◽  
Explicit Bounds ◽  
A Finite Group

A classical theorem of Jordan and Schur states that if G is a finite group of s × s matrices over a field K whose characteristic does not divide |G|, then G has an abelian subgroup of index bounded by a function of s. There are several direct and elegant proofs of this, leading to explicit bounds (4), (18).

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