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Let G be a group and denote its Frattini subgroup by Φ(G). I know that Φ(G) is the intersection of the maximal subgroups of G, and I know that it is the set of 'non-generators'

Can we compute frattini sub group for any group ? For example we can find frattini sub group for any cyclic group . What is history of frattini subgroup ? Why frattini sub group is important?

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There is no short answer for all your questions. Can one compute the Frattini subgroup? Yes, there are algorithms for doing it. I will not go into detail, but just give a reference by Bettina Eick. For cyclic groups one should know the Frattini subgroup. For example, $\Phi(C_p)=1$ for primes $p$ and $\Phi(C_{p^a})\cong C_{p^{a-1}}$. Other examples examples are: $$ \Phi(S_n)=1,\; \Phi(D_n)=1,\; \Phi (C_{p} \wr C_{p})\cong [C_{p} \wr C_{p},C_{p} \wr C_{p}] $$

Why are Fratini groups important? See the following references:

Intuition behind the Frattini subgroup

Some questions on the Frattini subgroup

Does every finite nilpotent group occur as a Frattini subgroup?

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