Suppose $G$ is an Abelian group of order $25$ and every element of $G$ satisfies the equation $x^{25}=e$. Prove or disprove that $G$ is cyclic I'm not confident in my solution so I was hoping to get insight into where I may have gone wrong with it. If anyone has some advice or a corrective suggestion please share.
Since the order of any element must divide $25$, the possibilities of $|x| = 1, 5, 25$. Since $x\neq e, |x| \neq 1.$ If $|x|=25$, then $G$ is cyclic and we are done. Therefore we assume $|x|=5$. Since the order of G is $25$, this implies that there are $24$ nonidentity elements and since $\phi(5)= 4$ which is a multiple of $24$, $|x| = 5$. Further, since $\gcd(25,5)\neq 1$, $G$ is not a cyclic group.
 A: Well, $x^{|G|}=e$ holds for every element of every finite group, so your question amounts to asking whether is there any noncyclic abelian group of order $25$. And the other answers have already shown that there is one.
A: This in your OP is not a proof, as it does not show existence of an Abelian group $G$ with $|G|=25$ and $G$ not cyclic. [In fact, every group with $25$ elements has precisely $24$ non-identity elements.] You do conclude correctly that if such a group were to exist it would have an element $x$ with order exactly $5$, but that is it, and infact every group with $25$ elements has an element of order exactly $5$.]
There does exist a group $G$ however, that has exactly $25$ elements and is abelian and not cyclic. Consider $G=\mathbb{Z}/5 \mathbb{Z} \times \mathbb{Z}/5 \mathbb{Z}$, where the operation is component-wise addition. Then the resulting group has order $25$, is abelian, and is not cyclic. Indeed, you can check that every non-identity element in $G$ has order $5$.
A: By the structure theorem we have that there's only $2$ possibilities: $$\Bbb Z_{25}$$ and $$\Bbb Z_5×\Bbb Z_5.$$
One is cyclic;  one isn't.
Note that in the cyclic one there's $\varphi (25)=20$ elements of order $25;$  and $\varphi (5)=4$ of order $5.$
And don't forget what @lulu said. For the order of any element divides the order of the group.
