# Math Help - Abelian Group 2.12

1. ## Abelian Group 2.12

Let {G;*} be a group such that a^2 = e for every a belonging to G. Prove that G must be a abelian group.

2. Originally Posted by tigergirl
Let {G;*} be a group such that a^2 = e for every a belonging to G. Prove that G must be a abelian group.
$
a.b = b.a.a^{-1}.b^{-1}.a.b$

$= b.a.a^{-1}.b^{-1}.(b.a).(b.a).a.b$
$= b.a.e.e$

3. $a^2=e$
$b^2=e$
$abab=(ab)(ab)=(ab)^2=e$
Now take this and multiply on the left by a and right by b.

$a(abab)b=a(e)b$
$(aa)ba(bb)=ab$
$ba=ab$