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

Statement

For every finite $p$ -group $P$ , the subgroup $P^{p}$ is characteristic and

$Φ (P) = P^{'} P^{p},$

where $P^{'} = [P, P]$ (Commutators $[g, h] = g h g^{- 1} h^{- 1}$ and the commutator subgroup $[G, G]$ ) and $P^{p}$ is the subgroup generated by the $p$ th powers.

Facts & Assumptions

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

[F1]

For a group $G$ and a prime $p$ , the $p$ th-power subgroup is $G^{p} = ⟨ g^{p} : g \in G ⟩$ (The $p$ th-power subgroup $G^{p}$ ).

[L1]

For a finite $p$ -group $P$ , $P / Φ (P)$ is elementary abelian, and $P / N$ is elementary abelian exactly when $Φ (P) \leq N$ (The Frattini quotient is the largest elementary abelian quotient of a finite $p$ -group).

[L2]

The commutator subgroup $P^{'}$ is normal in $P$ (The commutator subgroup is normal).

[L3]

If $H \leq G$ and $N ⊴ G$ , then $H N$ is a subgroup (If $H \leq G$ and $N ⊴ G$ , then $H N$ is a subgroup and $H \cap N ⊴ H$ ).

Proof

technique · direct

1.1

Every automorphism sends $g^{p}$ to $α (g)^{p}$ , so it preserves the generating set in [F1] and hence $P^{p}$ is characteristic; conjugation by an element of $P$ is an automorphism, so $P^{p}$ is normal. By [L2], $P^{'}$ is normal as well, so [L3] makes $K : = P^{'} P^{p}$ a subgroup, and $g K g^{- 1} = (g P^{'} g^{- 1}) (g P^{p} g^{- 1}) = P^{'} P^{p} = K$ for every $g \in P$ makes it normal.

givenF1L2L3algebra

2.1

The elementary abelian quotient $P / Φ (P)$ is abelian and has exponent $p$ by [L1], so it kills every commutator and every $p$ th power. Thus $K = P^{'} P^{p} \leq Φ (P)$ .

step 1.1F1L1algebra

3.1

The quotient $P / K$ is abelian because it kills $P^{'}$ , and every element has $p$ th power one because it kills $P^{p}$ . It is therefore elementary abelian, so the kernel criterion in [L1] gives $Φ (P) \leq K$ . Together with step 2.1 this proves equality.

step 1.1step 2.1L1algebra∎

Depends on

Used by

Φ(P)=P² for a finite 2-group Corollary
The 3×3 upper-unitriangular group over a prime field has generator rank two Example
The Frattini subgroup of a nontrivial cyclic p-group Example
If H≤ P are finite p-groups, then Φ(H)≤Φ(P) Proposition
Φ(P/N)=Φ(P)N/N for a normal subgroup of a finite p-group Proposition
Φ(P× Q)=Φ(P)×Φ(Q) for finite p-groups Proposition

Dependency tree · two levels

25 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

D. A. Craven, The Theory of p-Groups, Proposition 2.25 (standard reference, not scraped)
M. van Beek, Topics in Finite p-Groups, Proposition 3.5 (standard reference, not scraped)

Φ(P)=P′PpΦ(P)=P′Pp for a finite pp-group