The sub-class sizes of some elements being square free

2021 ◽  
Author(s):  
Xianhe Zhao ◽  
Yanyan Zhou ◽  
Ruifang Chen ◽  
Qin Huang
Keyword(s):  
Finite Group ◽  
Conjugacy Class ◽  
Finite Groups ◽  
Prime Power ◽  
Prime Factor ◽  
Subnormal Subgroup ◽  
Prime Power Order ◽  
Conjugacy Class Sizes ◽  
A Finite Group

Let [Formula: see text] be an element of a finite group [Formula: see text], and [Formula: see text] a prime factor of the order of [Formula: see text]. It is clear that there always exists a unique minimal subnormal subgroup containing [Formula: see text], say [Formula: see text]. We call the conjugacy class of [Formula: see text] in [Formula: see text] the sub-class of [Formula: see text] in [Formula: see text], see [G. Qian and Y. Yang, On sub-class sizes of finite groups, J. Aust. Math. Soc. (2020) 402–411]. In this paper, assume that [Formula: see text] is the product of the subgroups [Formula: see text] and [Formula: see text], we investigate the solvability, [Formula: see text]-nilpotence and supersolvability of the group [Formula: see text] under the condition that the sub-class sizes of prime power order elements in [Formula: see text] are [Formula: see text] free, [Formula: see text] free and square free, respectively, so that some known results relevant to conjugacy class sizes are generalized.

ON CONJUGACY CLASS SIZES AND CHARACTER DEGREES OF FINITE GROUPS

2013 ◽  
Vol 13 (02) ◽  
pp. 1350100 ◽  
Author(s):  
GUOHUA QIAN ◽  
YANMING WANG
Keyword(s):  
Finite Group ◽  
Conjugacy Class ◽  
Nilpotent Group ◽  
Finite Groups ◽  
Prime Power ◽  
P Element ◽  
Character Degrees ◽  
Prime Power Order ◽  
Conjugacy Class Sizes ◽  
A Finite Group

Let p be a fixed prime, G a finite group and P a Sylow p-subgroup of G. The main results of this paper are as follows: (1) If gcd (p-1, |G|) = 1 and p2 does not divide |xG| for any p′-element x of prime power order, then G is a solvable p-nilpotent group and a Sylow p-subgroup of G/Op(G) is elementary abelian. (2) Suppose that G is p-solvable. If pp-1 does not divide |xG| for any element x of prime power order, then the p-length of G is at most one. (3) Suppose that G is p-solvable. If pp-1 does not divide χ(1) for any χ ∈ Irr (G), then both the p-length and p′-length of G are at most 2.

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 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 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 Influence of Certain Abelian Subgroups on the Structure of Finite Groups

2008 ◽  
Vol 15 (03) ◽  
pp. 479-484 ◽  
Author(s):  
M. Ramadan
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Prime Power ◽  
Prime Power Order ◽  
Group A ◽  
Abelian Subgroups ◽  
A Finite Group

Let G be a finite group. A subgroup K of a group G is called an [Formula: see text]-subgroup of G if NG(K) ∩ Kx ≤ K for all x ∈ G. The set of all [Formula: see text]-subgroups of G is denoted by [Formula: see text]. In this paper, we investigate the structure of a group G under the assumption that certain abelian subgroups of prime power order belong to [Formula: see text].

ON -PARTS OF CONJUGACY CLASS SIZES OF FINITE GROUPS

2018 ◽  
Vol 97 (3) ◽  
pp. 406-411 ◽  
Author(s):  
YONG YANG ◽  
GUOHUA QIAN
Keyword(s):  
Finite Group ◽  
Conjugacy Class ◽  
Finite Groups ◽  
Conjugacy Classes ◽  
Conjugacy Class Sizes ◽  
A Finite Group

Let $G$ be a finite group. Let $\operatorname{cl}(G)$ be the set of conjugacy classes of $G$ and let $\operatorname{ecl}_{p}(G)$ be the largest integer such that $p^{\operatorname{ecl}_{p}(G)}$ divides $|C|$ for some $C\in \operatorname{cl}(G)$. We prove the following results. If $\operatorname{ecl}_{p}(G)=1$, then $|G:F(G)|_{p}\leq p^{4}$ if $p\geq 3$. Moreover, if $G$ is solvable, then $|G:F(G)|_{p}\leq p^{2}$.

A note on 𝑝-parts of conjugacy class sizes of finite groups

2019 ◽  
Vol 22 (5) ◽  
pp. 933-940
Author(s):  
Jinbao Li ◽  
Yong Yang
Keyword(s):  
Finite Group ◽  
Conjugacy Class ◽  
Finite Groups ◽  
Conjugacy Classes ◽  
Conjugacy Class Sizes ◽  
A Finite Group

Abstract Let G be a finite group and p a prime. Let {\operatorname{cl}(G)} be the set of conjugacy classes of G, and let {\operatorname{ecl}_{p}(G)} be the largest integer such that {p^{\operatorname{ecl}_{p}(G)}} divides {|C|} for some {C\in\operatorname{cl}(G)} . We show that if {p\geq 3} and {\operatorname{ecl}_{p}(G)=1} , then {\lvert G\mskip 1.0mu \mathord{:}\mskip 1.0mu O_{p}(G)\rvert_{p}\leq p^{3}} . This improves the main result of Y. Yang and G. Qian, On p-parts of conjugacy class sizes of finite groups, Bull. Aust. Math. Soc. 97 2018, 3, 406–411.

scholarly journals Finite Groups Whose Certain Subgroups of Prime Power Order Are -Semipermutable

ISRN Algebra ◽  
2011 ◽  
Vol 2011 ◽  
pp. 1-8
Author(s):  
Mustafa Obaid
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Sylow Subgroup ◽  
Prime Power ◽  
Prime Power Order ◽  
Group A ◽  
Abelian Subgroups ◽  
A Finite Group

Let be a finite group. A subgroup of is said to be S-semipermutable in if permutes with every Sylow -subgroup of with . In this paper, we study the influence of S-permutability property of certain abelian subgroups of prime power order of a finite group on its structure.

scholarly journals PGL2(q) cannot be determined by its cs

Filomat ◽  
2020 ◽  
Vol 34 (5) ◽  
pp. 1713-1719
Author(s):  
Neda Ahanjideh
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Conjugacy Class ◽  
Prime Power ◽  
Simple Groups ◽  
Conjugacy Class Sizes ◽  
Direct Factor ◽  
Almost Simple Groups ◽  
Trivial Element ◽  
A Finite Group

For a finite group G, let Z(G) denote the center of G and cs*(G) be the set of non-trivial conjugacy class sizes of G. In this paper, we show that if G is a finite group such that for some odd prime power q ? 4, cs*(G) = cs*(PGL2(q)), then either G ? PGL2(q) X Z(G) or G contains a normal subgroup N and a non-trivial element t ? G such that N ? PSL2(q)X Z(G), t2 ? N and G = N. ?t?. This shows that the almost simple groups cannot be determined by their set of conjugacy class sizes (up to an abelian direct factor).

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