# prove H is an abelian group

• Nov 5th 2009, 06:51 PM
apple2009
prove H is an abelian group
g²=I for all h is an element of H.
a)Prove: H is an abelian group
b)Prove: Suppose |H|<∞. Let {h₁,h₂,…..hn} be minimal set of generators for H. then http://www.mathhelpforum.com/math-he...2647988b-1.gif
• Nov 5th 2009, 07:17 PM
Drexel28
Quote:

Originally Posted by apple2009
g²=I for all h is an element of H.
a)Prove: H is an abelian group
b)Prove: Suppose |H|<∞. Let {h₁,h₂,…..hn} be minimal set of generators for H. The |H|=2^n

Problem: Suppose $G$ is a group such that $g^2=e_G\quad\forall g\in G$. Prove that $G$ is abelian.

Proof (1): Note that $g^2=e\implies g=g^{-1}$. Therefore $\left(ab\right)=\left(ab\right)^{-1}=b^{-1}a^{-1}=ba$.

Proof (2): Using the above we can see that $ab=a\left(e_G\right)b=a(ab)^2b=(aa)ba(bb)=a^2bab^2 =ba$

Part b doesn't make sense. Is there a typo?
• Nov 5th 2009, 07:41 PM
Drexel28
Quote:

Originally Posted by apple2009
g²=I for all h is an element of H.
a)Prove: H is an abelian group
b)Prove: Suppose |H|<∞. Let {h₁,h₂,…..hn} be minimal set of generators for H. The |H|=2^n

I think it should be "then $\left|H\right|=2^n$. If so, I think you ought to be able to make a contrived answer from the way the question is posed. What is significant looking about that expression?