Prove that a group is abelian. [duplicate]

Question

Possible Duplicate:
Group where every element is order 2

Let $(G,\star)$ be a group with identity element $e$ such that $a \star a = e$ for all $a \in G$. Prove that $G$ is abelian.

Ok, what i got is this: we want to prove that a*b=b*a, i.e. if a*a=e , a=a' where a' is the inverse and b*b=e, b=b' where b' is the inverse so a*b=(a*b)'=b'*a'=b*a....

Answer 1

HINT: You need to prove $ab = ba$, $\forall a,b \in G$. Note that since $ab \in G$ we also have that $(ab)^2 = e$ i.e. $(ab)(ab) = e$. Further, $a^{-1} = a$ and $b^{-1} = b$. Can you now finish this off?

Answer 2

If $a*a=e$ then $a=a^{-1}$. It follows that $(a*b)^{-1}=a*b$ but $a*b= b^{-1}*a^{-1}$ you get the desired equality for comparing both expressions.

Answer 3

Given $a,b\in G$ we want to know $ab=ba$, i.e. $aba^{-1}b^{-1}=e$.

What we know is that $a*a=e$, i.e. $a=a^{-1}$, similarly $b=b^{-1}$, so what we need to verify is that $abab=e$.

Answer 4

There are groups with $a\cdot a = e$ for all $a\in G$ with more than one element, so that reasoning will not work.

What you may want to think about: If $a\cdot a = e$ for all $a\in G$, then what does this tell you about the element $ab$? We know $(ab)^2 = e$ and so...