$p$-groups have normal subgroups of each order
8841
A $p$-group of order $p^n$ has a normal subgroup of order $p^k$ for each $0\le k \le n$
4883
Let $H$ be a subgroup of a group $G$ such that $x^2 \in H$ , $\forall x\in G$ . Prove that $H$ is a normal subgroup of $G$
67272
Deduce that if $G$ is a finite $p$-group, the number of subgroups of $G$ that are not normal is divisible by $p$
68201
Cyclic group of order $n$
70373
Intersection of normal subgroups proof
123796
Order of automorphism group of a $p$-group is divisible by $p$.
145511
Suppose $H$ is the only subgroup of order $o(H)$ in the finite group $G$. Prove that $H$ is a normal subgroup of $G$.
142894
Showing that a group of order $pq$ is cyclic if it has normal subgroups of order $p$ and $q$
83710
Order of a homomorphism of groups
19720
A normal subgroup intersects the center of the $p$-group nontrivially
39938
Subgroups of a cyclic group and their order.
91287
Is $G/pG$ is a $p$-group?
118337
Let $S$ be a normal p-subgroup of a finite group $G$. Prove that $S \subseteq P$ for every Sylow p-subgroup $P$ of $G$
15526
Question on groups of order $pq$
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 $G$.
25404
Let $G$ be a group of order $2m$ where $m$ is odd. Prove that $G$ contains a normal subgroup of order $m$
11731
Normal subgroups of $S_4$