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

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.

The Influence of Certain Abelian Subgroups on the Structure of Finite Groups

Algebra Colloquium ◽

10.1142/s1005386708000461 ◽

2008 ◽

Vol 15 (03) ◽

pp. 479-484 ◽

Cited By ~ 3

Author(s):

M. Ramadan

Keyword(s):

Finite Group ◽

Finite Groups ◽

Prime Power ◽

Prime Power Order ◽

Group A ◽

Abelian Subgroups ◽

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].

SS-QUASINORMAL SUBGROUPS OF FINITE GROUPS

10.1142/s0219498810004336 ◽

2010 ◽

Vol 09 (06) ◽

pp. 977-984 ◽

Cited By ~ 2

Author(s):

TAO ZHAO ◽

XIANHUA LI

Keyword(s):

Finite Group ◽

Finite Groups ◽

Sylow Subgroup ◽

Prime Power ◽

Prime Power Order ◽

A subgroup H of a finite group G is said to be SS-quasinormal in G if there exists a supplement B of H in G such that H is permutable with every Sylow subgroup of B. In this paper, we get some new characterizations of supersolvability and p-nilpotency of G by assuming some subgroups of prime power order of G are SS-quasinormal.

On weakly ℋC-embedded subgroups of finite groups

10.1142/s0219498816500778 ◽

2016 ◽

Vol 15 (05) ◽

pp. 1650077 ◽

Cited By ~ 3

Author(s):

M. Asaad ◽

M. Ramadan

Keyword(s):

Finite Group ◽

Normal Subgroup ◽

Finite Groups ◽

Prime Power ◽

Prime Power Order ◽

Group A ◽

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.

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

Algebra Colloquium ◽

10.1142/s1005386711000538 ◽

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 ◽

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.

Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics ◽

Automorphism orbits of finite groups

10.1017/s1446788700027221 ◽

1986 ◽

Vol 40 (2) ◽

pp. 253-260 ◽

Cited By ~ 9

Author(s):

Thomas J. Laffey ◽

Desmond MacHale

Keyword(s):

Finite Group ◽

Automorphism Group ◽

Finite Groups ◽

Prime Power ◽

Structure Theorem ◽

Equivalence Classes ◽

Prime Power Order ◽

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.

Finite Groups with Minimal CSS-subgroups

European Journal of Pure and Applied Mathematics ◽

10.29020/nybg.ejpam.v14i3.4036 ◽

2021 ◽

Vol 14 (3) ◽

pp. 1002-1014

Author(s):

A. A. Heliel ◽

R. A. Hijazi ◽

S. M. Al-Shammari

Keyword(s):

Finite Group ◽

Normal Subgroup ◽

Finite Groups ◽

Sylow Subgroup ◽

Group A ◽

Let G be a finite group. A subgroup H of G is called SS-quasinormal in G if there is a supplement B of H to G such that H permutes with every Sylow subgroup of B. A subgroup H of G is called CSS-subgroup in G if there exists a normal subgroup K of G such that G = HK and H âˆ©K is SS-quasinormal in G. In this paper, we investigate the influence of minimal CSS-subgroups of G on its structure. Our results improve and generalize several recent results in the literature.

ON CONJUGACY CLASS SIZES AND CHARACTER DEGREES OF FINITE GROUPS

10.1142/s0219498813501004 ◽

2013 ◽

Vol 13 (02) ◽

pp. 1350100 ◽

Cited By ~ 4

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 ◽

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.

Finite Groups with Some -Supplemented Subgroups

Algebra ◽

10.1155/2013/153691 ◽

2013 ◽

Vol 2013 ◽

pp. 1-5

Author(s):

Guo Zhong ◽

Liying Yang ◽

Huaquan Wei ◽

Xuanlong Ma ◽

Jiayi Xia

Keyword(s):

Finite Group ◽

Finite Groups ◽

Sylow Subgroup ◽

Maximal Subgroups ◽

Group A ◽

Let be a subgroup of a finite group , a prime dividing the order of , and a Sylow -subgroup of for prime We say that is -supplemented in if there is a subgroup of such that and where denotes the subgroup of generated by all those subgroups of which are -quasinormally embedded in In this paper, we characterize -nilpotency and supersolvability of under the assumption that all maximal subgroups of are -supplemented in .

The sub-class sizes of some elements being square free

10.1142/s0219498823500743 ◽

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 ◽

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.

FINITE GROUPS WHOSE NONCENTRAL COMMUTING ELEMENTS HAVE CENTRALIZERS OF EQUAL SIZE

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972710000298 ◽

2010 ◽

Vol 82 (2) ◽

pp. 293-304 ◽

Cited By ~ 8

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 ◽

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.