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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?

amir bahadory
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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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Dietrich Burde
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  • Can you give more details of the reference by Bettina Eick. I am not able to open the link you attached. Thanks! – Jins Aug 03 '22 at 05:00
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    @Jins I can still open it. The title is " Special Presentations for Finite Soluble Groups and Computing (Pre)-Frattini Subgroups". You can google this for different links, e.g., here. – Dietrich Burde Aug 03 '22 at 07:55
  • Thanks, Dietrich Burde. I didnt get how the paper by Bettina Eick et al. presented algorithm for finding Frattini subgroup ? Is there any efficient algorithm known to find Frattini subgroup for general group ? or some special groups ? – Jins Aug 04 '22 at 04:51