On the Frattini subgroup of a polycyclic group

2021 ◽  
Author(s):  
Bettina Eick ◽  
Matthias Neumann-Brosig
Keyword(s):  
Polycyclic Group ◽  
Frattini Subgroup

scholarly journals Bounds in groups with trivial Frattini subgroup

2008 ◽  
Vol 319 (3) ◽  
pp. 893-896 ◽  
Author(s):  
Z. Halasi ◽  
K. Podoski
Keyword(s):  
Frattini Subgroup

scholarly journals A generalized Frattini subgroup of a finite group

1989 ◽  
Vol 12 (2) ◽  
pp. 263-266
Author(s):  
Prabir Bhattacharya ◽  
N. P. Mukherjee
Keyword(s):  
Finite Group ◽  
Maximal Subgroups ◽  
Frattini Subgroup ◽  
Definition Of ◽  
A Finite Group

For a finite group G and an arbitrary prime p, letSP(G)denote the intersection of all maximal subgroups M of G such that [G:M] is both composite and not divisible by p; if no such M exists we setSP(G)= G. Some properties of G are considered involvingSP(G). In particular, we obtain a characterization of G when each M in the definition ofSP(G)is nilpotent.

Addendum to "On the Frattini Subgroup"

1971 ◽  
Vol 27 (1) ◽  
pp. 63
Author(s):  
John Cossey ◽  
Alice Whittemore
Keyword(s):  
Frattini Subgroup

THE INFLUENCE OF Φ-S-SUPPLEMENTED SUBGROUPS ON THE STRUCTURE OF FINITE GROUPS

2012 ◽  
Vol 11 (04) ◽  
pp. 1250064
Author(s):  
CHANGWEN LI
Keyword(s):  
Finite Groups ◽  
Subnormal Subgroup ◽  
Frattini Subgroup

A subgroup H of a group G is called Φ-s-supplemented in G if there exists a subnormal subgroup K of G such that G = HK and H ∩ K ≤ Φ (H), where Φ(H) is the Frattini subgroup of H. We investigate the influence of Φ-s-supplemented subgroups on the p-nilpotency, p-supersolvability and supersolvability of finite groups.

On some polycyclic groups with small Hirsch length

2017 ◽  
Vol 16 (12) ◽  
pp. 1750237
Author(s):  
Heguo Liu ◽  
Fang Zhou ◽  
Tao Xu
Keyword(s):  
Abelian Subgroup ◽  
Polycyclic Group ◽  
Normal Abelian Subgroup ◽  
Finite Quotient ◽  
Polycyclic Groups

A polycyclic group [Formula: see text] is called an [Formula: see text]-group if every normal abelian subgroup of any finite quotient of [Formula: see text] is generated by [Formula: see text], or fewer, elements and [Formula: see text] is the least integer with this property. In this paper, the structure of [Formula: see text]-groups and [Formula: see text]-groups is determined.

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