scholarly journals Landau’s theorem on conjugacy classes for normal subgroups

2016 ◽  
Vol 26 (07) ◽  
pp. 1453-1466
Author(s):  
Antonio Beltrán ◽  
María José Felipe ◽  
Carmen Melchor
Keyword(s):  
Normal Subgroup ◽  
Finite Groups ◽  
Prime Power ◽  
Upper Bounds ◽  
Conjugacy Classes ◽  
Index Formula ◽  
Prime Power Order ◽  
Normal Subgroups ◽  
Small Index ◽  
Positive Integers

Landau’s theorem on conjugacy classes asserts that there are only finitely many finite groups, up to isomorphism, with exactly [Formula: see text] conjugacy classes for any positive integer [Formula: see text]. We show that, for any positive integers [Formula: see text] and [Formula: see text], there exist finitely many finite groups [Formula: see text], up to isomorphism, having a normal subgroup [Formula: see text] of index [Formula: see text] which contains exactly [Formula: see text] non-central [Formula: see text]-conjugacy classes. Upper bounds for the orders of [Formula: see text] and [Formula: see text] are obtained; we use these bounds to classify all finite groups with normal subgroups having a small index and few [Formula: see text]-classes. We also study the related problems when we consider only the set of [Formula: see text]-classes of prime-power order elements contained in a normal subgroup.

On S-Quasinormal and C-Normal Subgroups of Prime Power Order in Finite Groups

2011 ◽  
Vol 18 (04) ◽  
pp. 685-692
Author(s):  
Xuanli He ◽  
Shirong Li ◽  
Xiaochun Liu
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Prime Power ◽  
Maximal Subgroups ◽  
Prime Power Order ◽  
Normal Subgroups ◽  
A Finite Group

Let G be a finite group, p the smallest prime dividing the order of G, and P a Sylow p-subgroup of G with the smallest generator number d. Consider a set [Formula: see text] of maximal subgroups of P such that [Formula: see text]. It is shown that if every member [Formula: see text] of is either S-quasinormally embedded or C-normal in G, then G is p-nilpotent. As its applications, some further results are obtained.

scholarly journals On c-normality of finite groups

2005 ◽  
Vol 78 (3) ◽  
pp. 297-304 ◽  
Author(s):  
M. Asaad ◽  
M. Ezzat Mohamed
Keyword(s):  
Normal Subgroup ◽  
Finite Groups ◽  
Prime Power ◽  
Prime Power Order

AbstractA subgroup H of a finite G is said to be c-normal in G if there exists a normal subgroup N of G such that G = HN with H ∩ N ≤ HG = CoreG(H). We are interested in studying the influence of the c–normality of certain subgroups of prime power order on the structure of finite groups.

Finite Groups with Some Subgroups of Given Order

2013 ◽  
Vol 20 (03) ◽  
pp. 457-462 ◽  
Author(s):  
Jiangtao Shi ◽  
Cui Zhang ◽  
Dengfeng Liang
Keyword(s):  
Finite Groups ◽  
Prime Power ◽  
Solvable Groups ◽  
Complete Classification ◽  
Conjugacy Classes ◽  
Prime Power Order

Let [Formula: see text] be the class of groups of non-prime-power order or the class of groups of prime-power order. In this paper we give a complete classification of finite non-solvable groups with a quite small number of conjugacy classes of [Formula: see text]-subgroups or classes of [Formula: see text]-subgroups of the same order.

scholarly journals FINITE GROUPS WHOSE NONCENTRAL COMMUTING ELEMENTS HAVE CENTRALIZERS OF EQUAL SIZE

2010 ◽  
Vol 82 (2) ◽  
pp. 293-304 ◽  
Author(s):  
SILVIO DOLFI ◽  
MARCEL HERZOG ◽  
ENRICO JABARA
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Prime Power ◽  
Upper Bounds ◽  
Equal Size ◽  
Prime Power Order ◽  
Derived Length ◽  
A Finite Group

AbstractA finite group is called a CH-group if for every x,y∈G∖Z(G), xy=yx implies that $\|\cent Gx\| = \|\cent Gy\|$. Applying results of Schmidt [‘Zentralisatorverbände endlicher Gruppen’, Rend. Sem. Mat. Univ. Padova44 (1970), 97–131] and Rebmann [‘F-Gruppen’, Arch. Math. 22 (1971), 225–230] concerning CA-groups and F-groups, the structure of CH-groups is determined, up to that of CH-groups of prime-power order. Upper bounds are found for the derived length of nilpotent and solvable CH-groups.

A note on class sizes of vanishing elements in finite groups

2021 ◽  
pp. 2250067
Author(s):  
Qingjun Kong ◽  
Shi Chen
Keyword(s):  
Finite Group ◽  
Conjugacy Class ◽  
Finite Groups ◽  
Prime Power ◽  
Prime Power Order ◽  
Normal Subgroups ◽  
Conjugacy Class Sizes ◽  
Vanishing Elements ◽  
A Finite Group

Let [Formula: see text] and [Formula: see text] be normal subgroups of a finite group [Formula: see text]. We obtain th supersolvability of a factorized group [Formula: see text], given that the conjugacy class sizes of vanishing elements of prime-power order in [Formula: see text] and [Formula: see text] are square-free.

On weakly ℋC-embedded subgroups of finite groups

2016 ◽  
Vol 15 (05) ◽  
pp. 1650077 ◽  
Author(s):  
M. Asaad ◽  
M. Ramadan
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Prime Power ◽  
Prime Power Order ◽  
Group A ◽  
A Finite Group

Let [Formula: see text] be a finite group. A subgroup [Formula: see text] of [Formula: see text] is said to be an [Formula: see text]-subgroup of [Formula: see text] if there exists a normal subgroup [Formula: see text] of [Formula: see text] such that [Formula: see text] and [Formula: see text] for all [Formula: see text] We say that [Formula: see text] is weakly [Formula: see text]-embedded in [Formula: see text] if there exists a normal subgroup [Formula: see text] of [Formula: see text] such that [Formula: see text] and [Formula: see text] for all [Formula: see text]. In this paper, we investigate the structure of the finite group [Formula: see text] under the assumption that some subgroups of prime power order are weakly [Formula: see text]-embedded in [Formula: see text]. Our results improve and generalize several recent results in the literature.

scholarly journals Elements of prime power order and their conjugacy classes in finite groups

2005 ◽  
Vol 78 (2) ◽  
pp. 291-295 ◽  
Author(s):  
László Héthelyi ◽  
Burkhard Külshammer
Keyword(s):  
Positive Integer ◽  
Finite Groups ◽  
Prime Power ◽  
Conjugacy Classes ◽  
Simple Groups ◽  
Prime Power Order ◽  
Finite Simple Groups

AbstractWe show that, for any positive integer k, there are only finitely many finite groups, up to isomorphism, with exactly k conjugacy classes of elements of prime power order. This generalizes a result of E. Landau from 1903. The proof of our generalization makes use of the classification of finite simple groups.

scholarly journals Automorphism orbits of finite groups

1986 ◽  
Vol 40 (2) ◽  
pp. 253-260 ◽  
Author(s):  
Thomas J. Laffey ◽  
Desmond MacHale
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Finite Groups ◽  
Prime Power ◽  
Structure Theorem ◽  
Equivalence Classes ◽  
Prime Power Order ◽  
A Finite Group

AbstractLet G be a finite group and let Aut(G) be its automorphism group. Then G is called a k-orbit group if G has k orbits (equivalence classes) under the action of Aut(G). (For g, hG, we have g ~ h if ga = h for some Aut(G).) It is shown that if G is a k-orbit group, then kGp + 1, where p is the least prime dividing the order of G. The 3-orbit groups which are not of prime-power order are classified. It is shown that A5 is the only insoluble 4-orbit group, and a structure theorem is proved about soluble 4-orbit groups.

The Laitinen Conjecture for finite non-solvable groups

2012 ◽  
Vol 56 (1) ◽  
pp. 303-336 ◽  
Author(s):  
Krzysztof Pawałowski ◽  
Toshio Sumi
Keyword(s):  
Finite Group ◽  
Fixed Points ◽  
Solvable Group ◽  
Prime Power ◽  
Solvable Groups ◽  
Conjugacy Classes ◽  
Prime Power Order ◽  
Algebraic Condition ◽  
Smooth Action

AbstractFor any finite group G, we impose an algebraic condition, the Gnil-coset condition, and prove that any finite Oliver group G satisfying the Gnil-coset condition has a smooth action on some sphere with isolated fixed points at which the tangent G-modules are not isomorphic to each other. Moreover, we prove that, for any finite non-solvable group G not isomorphic to Aut(A6) or PΣL(2, 27), the Gnil-coset condition holds if and only if rG ≥ 2, where rG is the number of real conjugacy classes of elements of G not of prime power order. As a conclusion, the Laitinen Conjecture holds for any finite non-solvable group not isomorphic to Aut(A6).

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