$G$ be a group of order $n$. $G$ has one and only one subgroup of order of every divisor of $n$. Then G is cyclic.
Is the statement true?
I know the statement "$G$ be an Abelian group of order $n$. $G$ has atmost one subgroup of order of every divisor of $n$. Then G is cyclic." is true.
Can anyone please confirm me whether my statement is true or not?
I have copied this answer from the answer of https://math.stackexchange.com/users/223481/censi-li
Let $\varphi$ denote the Euler function, then you constraint forces that $G$ has at most $\varphi(d)$ elements of order $d$ for each $d\mid n$. But $\sum_{d\mid n}\varphi(d)=n$, so $G$ must contain exactly $\varphi(d)$ elements of order $d$ for each $d\mid n$, in particular, $G$ contains an element of order $n$.
