# Characterization of subgroup dual to Frattini Subgroup

Let $G$ be a group and let $\mathcal{L}(G)$ denote the complete lattice of subgroups of $G$. We have that every automorphism of $G$ induces a lattice-automorphism on $\mathcal{L}(G)$. From here we see that every maximal subgroup of $G$ is sent to a maximal subgroup of $G$. Considering the infimum (or intersection) of all the maximal subgroups yields a characteristic subgroup of $G$ which is commonly called the Frattini subgroup. Dually, we can see that every minimal subgroup of $G$ is also sent to a minimal subgroup. Considering the supremum of all the minimal subgroups of $G$ yields another characteristic subgroup of $G$.

I have two questions. First, whether there are any references which consider this specific subgroup, as I haven't been able to find any literature on it.

Second and more pertinent, I was wondering if anyone could find a characterization of this subgroup which reflects the characterization of the Frattini subgroup. That is, the Frattini subgroup can be defined as the subgroup of all non-generators; does there exist a similar property for this other subgroup?

Letting $\Psi(G)$ represent this specific subgroup, I've found that if $A\subseteq G$ is another subset of $G$ such that $\Psi(G)\not\subseteq A$ and $\langle A, \Psi(G)\rangle = G$ then $\langle A\rangle\neq G$. But I'm looking for a characterization in terms of the elements of $\Psi(G)$.

• How would "the supremum of all minimal (or whatever) sbgps." be defined? And how is it characteristic? – DonAntonio Apr 8 '14 at 16:40
• A subgroup is minimal if and only if it has prime order, so this subgroup is generated by all elements of prime order. – Derek Holt Apr 8 '14 at 16:43
• @DonAntonio It is defined as the subgroup generated by all minimal subgroups (the supremum of the minimal subgroups in the subgroup lattice). It's characteristic as an automorphism of a group induces a lattice automorphism which permutes the minimal subgroups, thus the group generated by them does not change. – Robert Wolfe Apr 8 '14 at 16:54
• Ok. "Subgroup generated by some subgroups" is somethink I can chew. "The supremum of some subgroups" is something I am deeply ignorant of. Thanks, @Bryan – DonAntonio Apr 8 '14 at 16:55
• Apparently, this idea was first defined by Ito, and is mentioned in books.google.com/… – Justin Benfield Jun 29 '16 at 4:14