Showing that a group of prime order is in the center of a $p$-group.
138108
If a normal subgroup $N$ of order $p$($p$ prime) is contained in a group $G$ of order $p^n$,then $N$ is in the center of $G$.
115726
For $p,q$ prime and $p \neq q$ show that $C_p \times C_q \cong C_{pq}$
134820
$p$ prime, Group of order $p^n$ is cyclic iff it is an abelian group having a unique subgroup of order $p$
152481
If order of group is $p^2$, where $p$ is prime, how can you deduce $G$ is isomorphic to $C_{p^2}$ or $C_p \times C_p$?
113936
Showing that a finite abelian group has a subgroup of order m for each divisor m of n
33714
$p$ prime, $p\mid a^k \Rightarrow p^k\mid a^k$
38153
Looking for a simple proof that groups of order $2p$ are up to isomorphism $\mathbb{Z}_{2p}$ and $D_{p}$ .
61546
Prove that for each prime $p$ there exists a nonabelian group of order $p^3$
122970
A question on $p$-groups, and order of its commutator subgroup.
131204
A non-trivial normal subgroup N of a finite group G or order $p^n$ is such that $N\cap Z(G)\neq \{ 1\}$
104080
Group with order $pq$ has subgroups of order $p$ and $q$
7108
Normal subgroup of prime index
12502
The center of a non-Abelian group of order 8
156319
Show that a group with $|G| = 33$ contains an element of order $3$
17178
Order of $\phi(g)$ divides the order of $g$
33300
Showing order of $Z (G) =1$ or $pq$
97833
Prove that if $k\mid m,$ then $Z_m$ has a subgroup of order $k.$