代數學-請高手回答

Question

Prove that if G is an abelian group , written multiplicatively,with identity e,
then all elements x of G satisfying the equation x^2=e form a subgroup H of G.

Answer 1

H={ x | x in G and x² = e }

(1) e in H (因 e² = e )

(2) if x, y in H, then xy in H
因 (xy)² = xyxy = xxyy (abelian & associative)
= x² y² = ee = e

(3) if x in H, then x^(-1) in H
因 x² = e => x^(-1) = x
故 x^(-1) in H

由(1),(2),(3)知, H 為 G 之 subgroup