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Minimal generating sets of a finite $p$ -group have size $d (P)$

Statement

Every minimal generating set of a finite $p$ -group $P$ has size $d (P)$ , and every generating set contains a minimal generating subset of that size.

Facts & Assumptions

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

[L1]

A subset $X \subseteq P$ is minimally generating exactly when the quotient map restricts to a bijection from $X$ onto a basis of $P / Φ (P)$ (Burnside Basis Theorem).

[L2]

Every spanning subset of a finite elementary abelian $p$ -group contains a basis, and all bases have the same finite size (Finite elementary abelian $p$ -groups have bases, basis extension, and a well-defined dimension).

[F1]

The generator rank $d (P)$ is the common size of a basis of $P / Φ (P)$ (The generator rank $d (P)$ of a finite $p$ -group).

Proof

technique · direct

1.1

By [L1], every minimal generating set is in bijection with a quotient basis. All such bases have size $d (P)$ by [F1] and [L2].

givenL1L2F1

2.1

If $Y$ generates $P$ , its quotient image spans. Choose by [L2] a basis contained in that finite image and, for each basis vector, retain one element of $Y$ mapping to it. The resulting subset maps bijectively onto the basis, so [L1] makes it minimally generating; step 1.1 gives its size.

givenstep 1.1L1L2algebra∎

Depends on

Used by

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Dependency tree · two levels

10 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

K. Conrad, Generating Sets, consequences of Theorem 6.12 (standard reference, not scraped)
M. van Beek, Topics in Finite p-Groups, Theorem 3.7 (standard reference, not scraped)

Minimal generating sets of a finite pp-group have size d(P)d(P)