Frattini quotient

This article defines a quotient-defining function, viz., a rule that takes a group and outputs a unique quotient group
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Definition

Symbol-free definition

The Frattini quotient of a group is defined as the quotient of the group by its Frattini subgroup.

Definition with symbols

The Frattini quotient of a group $G$ is defined as the group $G/\Phi (G)$ where $\Phi (G)$ denotes the Frattini subgroup of $G$.

Group properties satisfied

1. The Frattini quotient of any group is a Frattini-free group, and a group occurs as a Frattini quotient if and only if it is Frattini-free. In other words, it equals its own Frattini quotient. That's because, by the correspondence theorem, the maximal subgroups inside the Frattini quotient are precisely in correspodence with maximal subgroups in the whole group.
2. For a finite p-group, the Frattini quotient is always an elementary abelian group. Hence, for a nilpotent group, the Frattini quotient is a product of elementary abelian p-groups for possibly different primes $p$, all dividing the order of the original group.

Facts

There is a close relation between a group and its Frattini quotient. If the Frattini subgroup is finitely generated, then a subset of the group is a generating set if and only if its image in the Frattini quotient is a generating set for the Frattini quotient. Further information: Frattini subgroup is finitely generated implies subset is generating set iff image in Frattini quotient is

In particular, if the Frattini subgroup is finitely generated, then the quotient by the Frattini subgroup being cyclic implies that the Frattini subgroup is cyclic.