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An elementary abelian $p$ -group has a canonical $F_{p}$ -vector-space structure

Statement

The rule $\overset{ˉ}{a} \cdot x = x^{a}$ gives every elementary abelian $p$ -group its canonical $F_{p}$ -vector-space structure, with the group operation as vector addition and the identity as zero.

Facts & Assumptions

Given: An elementary abelian $p$ -group $E$ , a residue class $\overset{ˉ}{a} \in Z / p$ , and $x, y \in E$ , with integer powers as in Powers $g^{n}$ : natural exponents in a monoid and integer exponents in a group, with $g^{0} = e$ .

[F1]

An elementary abelian $p$ -group is a finite abelian $p$ -group in which every nonidentity element has order $p$ ; the trivial group is permitted (Elementary abelian $p$ -groups).

[L1]

For every prime $p$ , addition and multiplication make $Z / p$ a field (For every prime $p$ , the two operations on $Z / p$ make it a field).

[L2]

The additive structure of $Z / p$ is an abelian group and multiplication distributes over addition (For every natural $n$ , $(Z / n, +)$ is an abelian group, multiplication is a commutative monoid operation, and both distributive laws hold).

[L3]

For integers $a, b$ , one has $x^{a + b} = x^{a} x^{b}$ and $(x^{a})^{b} = x^{a b}$ ; if $x y = y x$ , then $(x y)^{a} = x^{a} y^{a}$ (Exponent laws in a group: $g^{m + n} = g^{m} g^{n}$ and $(g^{m})^{n} = g^{m n}$ for all $m, n \in Z$ , and $(g h)^{n} = g^{n} h^{n}$ when $g$ and $h$ commute).

Proof

technique · direct

1.1

If $a \equiv b (m o d p)$ , then $a - b = k p$ for an integer $k$ . Since $x^{p} = e$ by [F1], the power laws in [L3] give $x^{a} = x^{b} (x^{p})^{k} = x^{b}$ . Thus $\overset{ˉ}{a} \cdot x : = x^{a}$ is independent of the representative.

givenF1L2L3algebra

2.1

The power laws in [L3] and commutativity give $(\overset{ˉ}{a} + \overset{ˉ}{b}) \cdot x = (\overset{ˉ}{a} \cdot x) (\overset{ˉ}{b} \cdot x)$ , $(\overset{ˉ}{a} \overset{ˉ}{b}) \cdot x = \overset{ˉ}{a} \cdot (\overset{ˉ}{b} \cdot x)$ , $\overset{ˉ}{a} \cdot (x y) = (\overset{ˉ}{a} \cdot x) (\overset{ˉ}{a} \cdot y)$ , $\overset{ˉ}{1} \cdot x = x$ , and $\overset{ˉ}{0} \cdot x = e$ . Together with [L1] and [L2], these are the vector-space axioms.

step 1.1F1L1L2L3algebra

3.1

The scalar structure uses the existing abelian group law and does not change its elements. In particular its additive group remains finite, abelian, and of exponent $p$ , including the zero-dimensional trivial case.

step 2.1F1L1algebra∎

Depends on

Used by

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Sources

K. Conrad, Generating Sets, §6 (standard reference, not scraped)

An elementary abelian pp-group has a canonical FpFp​-vector-space structure