Let $F$ be a free group. Prove that the subgroup generated by all nth powers, $\{x^n|x\in F \}$ is a normal subgroup of $F$.
131516
Subgroup generated by n-th powers of elements
67272
Deduce that if $G$ is a finite $p$-group, the number of subgroups of $G$ that are not normal is divisible by $p$
94571
How can we prove that $N_G(H)=H$ implies $N_{f(G)}(f(H))=f(H)$
57227
Maximal normal $\pi$-subgroups and torsion subgroups
90332
Prove that that $U(n)$ is an abelian group.
123677
Normal subgroups of infinite symmetric group
46260
Prove that a group generated by two elements of order $2$, $x$ and $y$, is isomorphic to $D_{2n}$, where $n = |xy|.$
58766
free groups: $F_X\cong F_Y\Rightarrow|X|=|Y|$
44942
A group $G$ with a subgroup $H$ of index $n$ has a normal subgroup $K\subset H$ whose index in $G$ divides $n!$
44152
If $G$ has only one subgroup of order $n$, then that Subgroup is Normal
8841
A $p$-group of order $p^n$ has a normal subgroup of order $p^k$ for each $0\le k \le n$
76962
Sylow $p$-subgroups of $GL_n(\mathbb{F}_p)$
28108
Proving the kernel is a normal subgroup.
95534
If $|G|=p^nq$, then $G$ contains a unique normal subgroup of index $q$
12589
Show that every subgroup of the quaternion group Q is a normal subgroup of Q
112157
If $g$ is the generator of a group $G$, order $n$, when is $g^k$ a generator?
51359
Prove that every subgroup of an abelian group is a normal subgroup.