Subgroups of Groups with Several Characterizations

Question

The Frattini subgroup is defined for arbitrary group in at least two ways:

• It is the intersection of all maximal subgroups.

• It is the set of all non-generators of the group (i.e. it contains those $x\in G$ such that if a $S\cup \{x\}$ generates $G$ then $S$ also generates $G$).

If we move in the world of $p$-groups, the Frattini subgroup has two more characterizations:

• It is the smallest normal subgroup such that quotient by it is elementary abelian $p$-group.

• It is the product $[G,G]G^p$, where $G^p$ is the subgroup of $G$ generated by $p$-th powers of all elements.

Question: What are the other subgroups of a group (or $p$-group) which has several characterizations?

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(Of course, "normal subgroups" have some characterizations).

Answer 1

The commutator subgroup is the subgroup generated by commutators or the smallest normal subgroup with abelian factor group.

In a finite group, for any prime $p$, $O^{p}(G)$ is the subgroup generated by all elements whose order is coprime to $p$, and is also the smallest normal subgroup whose factor group is a $p$-group. Similarly for $O^{p'}(G)$. And it generalizes to sets $\pi$ of primes.

Also (in a finite group) $O_p(G)$ is the largest normal $p$-subgroup and is the intersection of all Sylow $p$-subgroups.