H normal in G

May 2010
3
0
group G = Z_4 x Z_4

Let H be a subgroup of G of order 4. Explain why H is normal in G and why the quotient group of G by H is Abelian and of order 4.
 

Drexel28

MHF Hall of Honor
Nov 2009
4,563
1,566
Berkeley, California
group G = Z_4 x Z_4

Let H be a subgroup of G of order 4. Explain why H is normal in G and why the quotient group of G by H is Abelian and of order 4.
Hint:
It's normal because the product of abelian groups is abelian and every subgroup of an abelian group is abelian.

Hint:

\(\displaystyle |G/H|=[G:H]=4=2^2\). Aren't all groups of order \(\displaystyle p^2\) abelian? Why?
 

Bruno J.

MHF Hall of Honor
Jun 2009
1,266
498
Canada
Quotients of abelian groups are always abelian!
 

Drexel28

MHF Hall of Honor
Nov 2009
4,563
1,566
Berkeley, California
Quotients of abelian groups are always abelian!
We seem to be clashing on what the OP is most likely to know today, huh? haha

Of course this is true since the quotient group will be the image of the canonical homomorphism and the image of an abelian group under a homomorphism is always abelian.

The OP will have to decide which he likes better I guess.