Frattini-in-center group
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This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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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.
