Let $\widehat{F_2}$ be the free profinite group of rank 2. Is its Frattini subgroup $\Phi(\widehat{F_2})$ trivial?

I know that the Frattini subgroup of a pro-$p$ group is open. On the other hand, the Frattini subgroup of the abstract free group $F_2$ is trivial.

The general profinite case $\widehat{F_2}$ however seems to resist my googling attempts.

I believe $\Phi(\widehat{F_2})$ being trivial is equivalent to - Every finite 2-generated group $G$ is the quotient of a finite 2-generated group $G'$ with $\Phi(G') = 1$.

If $\Phi(\widehat{F_2})$ isn't trivial, is there a description of the Frattini quotient $\widehat{F_2}/\Phi(\widehat{F_2})$ ?


It is trivial. This is a special case of Corollary 8.7.5 of the book of Ribes and Zalesskii. It uses that every proper open normal subgroup of a closed normal subgroup of a free profinite group is free profinite and that the Frattini subgroup is pronilpotent.


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