Is the group of order 175 abelian?

[lmedin02]

Homework Statement

Prove that the group of order 175 is abelian.

Homework Equations

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.

 

[rochfor1]

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

 

[SiddharthM]

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