Let G be an abelian group of order 175 (=5*5*7). Assume $x^5=e$ has at least seven solutions. What is G isomorphic to?
I see and can show that G is isomorphic to the its Sylow subgroups (orders 7 and 25), and I know that finite abelian groups are direct products of cyclic groups. How can I combine these?
Can I say that the order 7 Sylow subgroup is isomorphic to a cyclic subgroup or order 7 (using that finite abelian groups are direct products of cyclic groups)?
And then is there a way to break the order 25 Sylow group into the product of 2 cyclic groups each of order 5 so that G is isomorphic to the product of an order 7 cyclic group and two order 5 subgroups?