Finite Groups with Nilpotent Subgroups of Even Order

2021 ◽  
Author(s):  
Yan Deng ◽  
Wei Meng ◽  
Jiakuan Lu
Keyword(s):  
Finite Groups ◽  
Even Order

scholarly journals Nonabelian Sylow subgroups of finite groups of even order

1998 ◽  
Vol 4 (12) ◽  
pp. 88-90
Author(s):  
Naoki Chigira ◽  
Nobuo Iiyori ◽  
Hiroyoshi Yamaki
Keyword(s):  
Finite Groups ◽  
Sylow Subgroups ◽  
Even Order

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].

Class-Preserving Coleman Automorphisms of Finite Groups with Prescribed Centralizers

2017 ◽  
Vol 24 (02) ◽  
pp. 351-360
Author(s):  
Zhengxing Li ◽  
Hongwei Gao
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Inner Automorphism ◽  
Coleman Automorphism ◽  
Even Order ◽  
A Finite Group

Let G be a finite group. It is proved that any class-preserving Coleman automorphism of G is an inner automorphism whenever G belongs to one of the following two classes of groups: (1) CN-groups, i.e., groups in which the centralizer of any element is nilpotent; (2) CIT-groups, i.e., groups of even order in which the centralizer of any involution is a 2-group. In particular, the normalizer conjecture holds for both CN-groups and CIT-groups. Additionally, some other results are also obtained.

Non-abelian Sylow subgroups of finite groups of even order

2000 ◽  
Vol 139 (3) ◽  
pp. 525-539 ◽  
Author(s):  
Naoki Chigira ◽  
Nobuo Iiyori ◽  
Hiroyoshi Yamaki
Keyword(s):  
Finite Groups ◽  
Sylow Subgroups ◽  
Even Order

scholarly journals Finite Groups with All Maximal Subgroups of Prime or Prime Square Index

1964 ◽  
Vol 16 ◽  
pp. 435-442 ◽  
Author(s):  
Joseph Kohler
Keyword(s):  
Finite Group ◽  
Maximal Subgroup ◽  
Finite Groups ◽  
Prime Order ◽  
Maximal Subgroups ◽  
Odd Order ◽  
Order Case ◽  
Chief Factors ◽  
Even Order ◽  
A Finite Group

In this paper finite groups with the property M, that every maximal subgroup has prime or prime square index, are investigated. A short but ingenious argument was given by P. Hall which showed that such groups are solvable.B. Huppert showed that a finite group with the property M, that every maximal subgroup has prime index, is supersolvable, i.e. the chief factors are of prime order. We prove here, as a corollary of a more precise result, that if G has property M and is of odd order, then the chief factors of G are of prime or prime square order. The even-order case is different. For every odd prime p and positive integer m we shall construct a group of order 2apb with property M which has a chief factor of order larger than m.

scholarly journals EXTENSION OF AUTOMORPHISMS OF SUBGROUPS

2012 ◽  
Vol 54 (2) ◽  
pp. 371-380
Author(s):  
G. G. BASTOS ◽  
E. JESPERS ◽  
S. O. JURIAANS ◽  
A. DE A. E SILVA
Keyword(s):  
Abelian Group ◽  
Finite Groups ◽  
Dihedral Group ◽  
Abelian Groups ◽  
Alternating Group ◽  
Finite Abelian Groups ◽  
Quaternion Group ◽  
Icosahedral Group ◽  
Odd Order ◽  
Even Order

AbstractLet G be a group such that, for any subgroup H of G, every automorphism of H can be extended to an automorphism of G. Such a group G is said to be of injective type. The finite abelian groups of injective type are precisely the quasi-injective groups. We prove that a finite non-abelian group G of injective type has even order. If, furthermore, G is also quasi-injective, then we prove that G = K × B, with B a quasi-injective abelian group of odd order and either K = Q8 (the quaternion group of order 8) or K = Dih(A), a dihedral group on a quasi-injective abelian group A of odd order coprime with the order of B. We give a description of the supersoluble finite groups of injective type whose Sylow 2-subgroup are abelian showing that these groups are, in general, not quasi-injective. In particular, the characterisation of such groups is reduced to that of finite 2-groups that are of injective type. We give several restrictions on the latter. We also show that the alternating group A5 is of injective type but that the binary icosahedral group SL(2, 5) is not.

scholarly journals On finite groups with prescribed two-generator subgroups and integral Cayley graphs

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Yan-Quan Feng ◽  
István Kovács
Keyword(s):  
Finite Groups ◽  
Cayley Graphs ◽  
Integral Spectrum ◽  
Even Order

Abstract In this paper, we characterize the finite groups 𝐺 of even order with the property that, for any involution 𝑥 and element 𝑦 of 𝐺, ⟨ x , y ⟩ \langle x,y\rangle is isomorphic to one of the following groups: Z 2 \mathbb{Z}_{2} , Z 2 2 \mathbb{Z}_{2}^{2} , Z 4 \mathbb{Z}_{4} , Z 6 \mathbb{Z}_{6} , Z 2 × Z 4 \mathbb{Z}_{2}\times\mathbb{Z}_{4} , Z 2 × Z 6 \mathbb{Z}_{2}\times\mathbb{Z}_{6} and A 4 A_{4} . As a result, a characterization will be obtained for the finite groups all of whose Cayley graphs of degree 3 have integral spectrum.

FINITE GROUPS ALL OF WHOSE MAXIMAL SUBGROUPS OF EVEN ORDER ARE ℋp-GROUPS

2014 ◽  
Vol 13 (05) ◽  
pp. 1350148
Author(s):  
WEI MENG ◽  
JIAKUAN LU
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Prime Order ◽  
Cyclic Subgroup ◽  
Maximal Subgroups ◽  
Group A ◽  
Even Order ◽  
A Finite Group

Let G be a finite group. A subgroup H of G is called an ℋ-subgroup of G if NG(H) ∩ Hg ≤ H for all g ∈ G; G is said to be an ℋp-group if every cyclic subgroup of G of prime order or order 4 is an ℋ-subgroup of G. In this paper, the structure of the finite groups all of whose maximal subgroups of even order are ℋp-subgroups have been characterized.

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