prove H is an abelian group

apple2009 · #1 $Old$ November 5th, 2009, 07:51 PM

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

Drexel28 · #2 $Old$ November 5th, 2009, 08:17 PM

Quote:

Originally Posted by apple2009 $View Post$

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?

Drexel28 · #3 $Old$ November 5th, 2009, 08:41 PM

Quote:

Originally Posted by apple2009 $View Post$

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?