Say I want to consider every distinct abelian group of order
up to isomorphism.
By the first Sylow theorem, I know that every group that has $p$ as a factor in its order, has a Sylow p-subgroup. I also know that a group of prime order is cyclic. (Correct so far?).
If I'm looking at a finitely generated group, say:
Then this group has elementary divisors $2,2,2,2, 3,3, 3,5^2$.
Does this mean that A only has Sylow 2-subgroups of order 2? Or does it have Sylow 2-subgroups of order $2^4$?
And am I right in that only the Sylow 2-subgroups of order 2 are cyclic, since 2 to any other power is not prime?
Thanks in advance!