Let p be prime. If a group has more than $p-1$ elements of order $p$, why can't the group be cyclic?
53207
Let $p$ be a prime. If a group has more than $p − 1$ elements of order $p$, why can’t the group be cyclic?
14890
Let $G=\langle a\rangle$ be a cyclic group of order $n$. Show that, for every divisor $d$ of $n$, there exists a subgroup of $G$ whose order is $d$.
104460
When a group is cyclic
39938
Subgroups of a cyclic group and their order.
129485
A subgroup of a cyclic group is cyclic - Understanding Proof
155054
$G$ be a group and $H$ be a normal subgroup of index $p$ ( a prime ) ; suppose $K$ is a subgroup of $G$ not contained in $H$ , then $G=HK$?
127306
Is the group of units of a finite ring cyclic?
101735
A finite $p$-group cannot be simple unless it has order $p$
36887
Suppose $G$ is a group of order 4. Show either $G$ is cyclic or $x^2=e$.
63499
Groups of prime squared order
118849
If a cyclic group has an element of infinite order, how many elements of finite order does it have?
99205
Group of order $p^k q$ where $p$ and $q$ are distinct primes - how to prove this property?
89382
In a group $G$ where $\exists!$ nontrivial, proper subgroup, show that $G$ is cyclic and $\lvert G\rvert=p^2$, for $p$ prime.
117206
$|G|$ and $|{\rm Aut}(G)|$ are relatively prime $\Rightarrow$ $G$ should be cyclic?
14003
$(\mathbb{Z}/2^n \mathbb{Z})^*$ is not cyclic Group for $n\geq 3$
61550
Existence of elements of order $p^k$ in a finite group
51196
A group of order $p^2q^2$ is never simple