Burnside basis theorem

Theorem 1

If $G$ is a finite $p$ -group, then $\\mathrm{Frat}\\,G=G^{\\prime}G^{p}$ , where $\\mathrm{Frat}\\,G$ is theFrattini subgroup, $G^{\\prime}$ the commutator subgroup, and $G^{p}$ is the subgroupgenerated by $p$ -th powers.

The theorem implies that $G/\\mathrm{Frat}\\,G$ is elementaryabelian, and thus has a minimal generating set ofexactly $n$ elements, where $|G:\\mathrm{Frat}\\,G|=p^{n}$ . Since any lift of such agenerating set also generates $G$ (by the non-generating property of theFrattini subgroup), the smallest generating set of $G$ alsohas $n$ elements.

The theorem also holds for profinite $p$ -groups (inverse limit of finite $p$ -groups).

单词	BurnsideBasisTheorem
释义	Burnside basis theorem Theorem 1 If $G$ is a finite $p$ -group, then $\\mathrm{Frat}\\,G=G^{\\prime}G^{p}$ , where $\\mathrm{Frat}\\,G$ is theFrattini subgroup, $G^{\\prime}$ the commutator subgroup, and $G^{p}$ is the subgroupgenerated by $p$ -th powers. The theorem implies that $G/\\mathrm{Frat}\\,G$ is elementaryabelian, and thus has a minimal generating set ofexactly $n$ elements, where $\|G:\\mathrm{Frat}\\,G\|=p^{n}$ . Since any lift of such agenerating set also generates $G$ (by the non-generating property of theFrattini subgroup), the smallest generating set of $G$ alsohas $n$ elements. The theorem also holds for profinite $p$ -groups (inverse limit of finite $p$ -groups).
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