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I suppose to have a 2-generated $p$-group $G$. I know that if $\langle a,b\rangle =G$, then $\langle aΦ(G), bΦ(G)\rangle = G/Φ(G)$, where $Φ(G)$ is the Frattini subgroup of $G$.

Is it also true that if I have $\langle cΦ(G),dΦ(G)\rangle = G/Φ(G)$, then $\langle c,d\rangle =G$ ?

J.-E. Pin
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Mary
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  • Since $\Phi(G)$ is the set of all non-generators, $S$ generates $G$ if and only if $\langle S,\Phi(G)\rangle$ generates $G$; hence from $\langle c,d,\Phi(G)\rangle = G$ you can deduce that $\langle c,d\rangle=G$. – Arturo Magidin Oct 09 '19 at 21:09

1 Answers1

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Yes, this is (not just for 2, but any number) the statement of Burnside’s basis theorem.

ahulpke
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