Homework Help: Classifying finite groups

1. Nov 8, 2009

lmedin02

1. The problem statement, all variables and given/known data
Prove that the group of order 175 is abelian.

2. Relevant equations

3. The attempt at a solution
|G|=175=527. Using the Sylow theorems it can be determined that G has only one Sylow 2-subgroup of order 25 called it H and only one Sylow 7-subgroups called it K. Thus, H and K are normal subgroups of G and G=H x K which is isomorphic to the direct product of H and K. Since |H|=52, then H is Abelian. Since K is of prime order then K is cyclic and therefore also Abelian.

I am not sure whether I can now conclude that G must be abelian since it is the external (or direct) product of abelian subgroups.

2. Nov 8, 2009

rochfor1

Can you try to prove that the direct product of abelian groups is abelian?

3. Jan 7, 2010

SiddharthM

direct products of abelian groups are abelian, this is obvious. Look at commutators and use the fact that they have trivial intersection.