Groups with a solvable subgroup of prime-power index

AbstractAll groups considered in this paper are finite. A subgroup $H$ of a group $G$ is called a primitive subgroup if it is a proper subgroup in the intersection of all subgroups of $G$ containing $H$ as a proper subgroup. He et al. [‘A note on primitive subgroups of finite groups’, Commun. Korean Math. Soc. 28(1) (2013), 55–62] proved that every primitive subgroup of $G$ has index a power of a prime if and only if $G/ \Phi (G)$ is a solvable PST-group. Let $\mathfrak{X}$ denote the class of groups $G$ all of whose primitive subgroups have prime power index. It is established here that a group $G$ is a solvable PST-group if and only if every subgroup of $G$ is an $\mathfrak{X}$-group.

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A note on elements of prime power index in finite groups

Monatshefte für Mathematik ◽

10.1007/s00605-015-0751-6 ◽

2015 ◽

Vol 179 (4) ◽

pp. 577-580

Author(s):

Qingjun Kong

Keyword(s):

Finite Groups ◽

Prime Power ◽

Power Index

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On Subgroups of Prime Power Index

Journal of the London Mathematical Society ◽

10.1112/s0024610700001125 ◽

2000 ◽

Vol 62 (2) ◽

pp. 407-422 ◽

Cited By ~ 1

Author(s):

Barbara Baumeister

Keyword(s):

Prime Power ◽

Power Index

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Finite groups with elements of prime power index

Journal of Algebra and Its Applications ◽

10.1142/s0219498815500954 ◽

2015 ◽

Vol 14 (06) ◽

pp. 1550095

Author(s):

Qingjun Kong

Keyword(s):

Finite Group ◽

Finite Groups ◽

Prime Power ◽

Power Index ◽

Prime Power Order ◽

A Finite Group

Let G be a finite group and let π be a set of primes. For an element x of G, let Ind G(x) denote the index of CG(x) in G. We prove that if Ind 〈a,x〉(x) is a π-number for every element a of prime power order in G, then Ind G(x) is a π-number.

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