4

I am new to algorithmic group theory. I have the following question:

Let $G$ be a group. The Frattini subgroup of $G$ is the intersection of all maximal subgroup of $G$, denoted by $\Phi(G)$. It is also equal to the set of all non-generating elements of $G$.

Question: Given a solvable group $G$ (by its multiplication table), and $H \leq G$, is there an algorithm to check if $H\leq \Phi(G)$?

Note: Suppose $K$ is $p$-group, then $\Phi(K)= K^{p}[K,K]$. Thus, given $H \leq K$ we can check if $H \leq \Phi(K)$? Hence we can solve the question for Nilpotent groups as well.

But, is there an algorithm that find Frattini subgroup for solvable groups or even super-solvable groups? or Can we direct answer if $H \leq \Phi(G)$ without finding $\Phi(G)$?

There are practical algorithm that finds the Frattini subgroup. I am interested in some theoretical algorithm that runs. Of course the trivial algorithm can finds subgroup lattice and find the Frattini subgroup explicitly. But is there an algorithm that runs in $|G|^c$ time for some constant $c$.

Pranjal
  • 299
  • 1
    As you point out yourself, the answer to your question is obviously yes. I would suggest that you edit your question and write "... is there a polynomial algorithm to check ...". In fact it would be equivalent to ask for a polynomial algorithm to compute (generators of) $\Phi(G)$. I don't know the answer immediately. In practical computation groups are not generally given by multiplication tables. – Derek Holt Sep 19 '23 at 08:26

0 Answers0