Subgroup(s) of a group of order 25 I am working on a problem (self-study) from Artin - 2.8.8 which goes:
"Let G be a group of order 25. Prove that G has at least one subgroup of order 5, and that if it contains only one subgroup of order 5, then it is a cyclic group."
I can see that there is an element of order 5, and this can generate a cyclic subgroup of order 5.
-- So my first question is: what would make one think that there might be more than one subgroup of order 5, and what would they look like. And what would be the criteria for there being only one rather than more than one?
For the second part, I am familiar with a proof that a group of order $p^2$ is abelian, by showing the center is all of G. 
-- My second question is how to show G is cyclic - and how does this use the stipulation that there is only one subgroup of order 5 (in the text there is a statement that if there is only one subgroup of a particular order then it is normal). And how does there being more than one subgroup of order 5 prevent G from being cyclic.
Thanks.

It is an easy exercise to show that if $c_d(G)$ denotes the number of cyclic subgroups of $G$ of order $d$ then $\displaystyle \sum_{d\mid |G|}c_d(G)\varphi(d)=|G|$ (just partition your group according to the elements orders). Now, if $G$ had only one subgroup of order $5$, then via the fact that all groups of order $5$ are cyclic we can conclude that $c_5(G)=1$. If then $c_{25}(G)=0$ then we'd see that
$$25=|G|=c_1(G)\varphi(1)+4c_5(G)+\varphi(5)=1+4=5$$
Of course, this is ridiculous. So, if $G$ contains only one subgroup of order $5$ then $c_d(G)\ne0$ and so there exists a cyclic subgroup of $G$ of order $25$ which, obviously, must be $G$ itself.
EDIT: Of course, there is nothing special about $5$. The above argument tells you that for a group of order $p^2$, $p$ prime, that being cyclic is equivalent to having one subgroup of order $p$. You will eventually learn that, up to isomorphism, the only groups of order $p^2$ are the cyclic one and the $2$-dimensional $\mathbb{F}_p$-space.
