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$Φ (P) = P^{2}$ for a finite $2$ -group

Statement

For every finite $2$ -group $P$ , $Φ (P) = P^{2}$ .

Facts & Assumptions

Given: A finite $2$ -group $P$ .

[L1]

For every finite $p$ -group $P$ , the subgroup $P^{p}$ is characteristic and $Φ (P) = P^{'} P^{p}$ ( $Φ (P) = P^{'} P^{p}$ for a finite $p$ -group).

[F1]

The square subgroup is $P^{2} = ⟨ g^{2} : g \in P ⟩$ (The $p$ th-power subgroup $G^{p}$ ).

[L2]

For $N ⊴ G$ , the quotient $G / N$ is abelian if and only if $G^{'} \leq N$ ( $G / N$ is abelian if and only if $[G, G] \subseteq N$ ).

Proof

technique · direct

1.1

By [L1], $P^{2}$ is characteristic and hence normal. Every element of $P / P^{2}$ has square one by [F1]. In any group of exponent at most two, $(x y)^{2} = e$ gives $x y = (x y)^{- 1} = y^{- 1} x^{- 1} = y x$ , so $P / P^{2}$ is abelian.

givenL1F1algebra

2.1

By [L2], step 1.1 gives $P^{'} \leq P^{2}$ . Substituting in [L1] yields $Φ (P) = P^{'} P^{2} = P^{2}$ , including $P = 1$ .

step 1.1L1F1L2algebra∎

Depends on

Used by

The Frattini subgroups of the dihedral and quaternion groups of order eight Example

Dependency tree · two levels

13 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

M. van Beek, Topics in Finite p-Groups, Lemma 3.6(iv) (standard reference, not scraped)

Φ(P)=P2Φ(P)=P2 for a finite 22-group