Frattini-in-center group: Difference between revisions
(New page: {{wikilocal}} {{group property}} ==Definition== A '''Frattini-in-center group''' is a group satisfying the following equivalent conditions: * Its [[defining ingredient::inner automo...) |
|||
| Line 16: | Line 16: | ||
* [[Weaker than::Special group]] | * [[Weaker than::Special group]] | ||
* [[Weaker than::Extraspecial group]] | * [[Weaker than::Extraspecial group]] | ||
* [[Weaker than::Group of Frattini class two]] | |||
===Weaker properties=== | ===Weaker properties=== | ||
* [[Stronger than::Group of | * [[Stronger than::Group of nilpotency class two]] | ||
==Facts== | ==Facts== | ||
Latest revision as of 17:11, 17 April 2010
BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
View a complete list of group properties
VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions
Definition
A Frattini-in-center group is a group satisfying the following equivalent conditions:
- Its inner automorphism group (i.e., the quotient group by its center) is Abelian and Frattini-free.
- Its commutator subgroup is contained in its Frattini subgroup, which in turn is contained in its center.
Relation with other properties
Stronger properties
Weaker properties
Facts
A group of prime power order is Frattini-in-center if and only if it satisfies the following equivalent conditions:
- It is a critical subgroup of itself.
- It can be realized as a critical subgroup of some group of prime power order.