Let $H$ be a subgroup of $G$. $S=\{g\in G\mid ghg^{-1}\in H\}$. Is $S= N_{G(H)}$?
41085
Conjugate subgroup strictly contained in the initial subgroup?
152071
Group $G$ is cyclic $\iff$ every subgroup of $G$ has the form $G^k$, where $G^k=\{g^k\mid g\in G\}$
90616
When is mapping $g$ to $g^{-1}$ a group homomorphism?
22477
If H is a subgroup of a group G and K is a normal subgroup of G the show that K is a normal subgroup of HK.
34724
For $G$ a group and $H\unlhd G$, then $G$ is solvable iff $H$ and $G/H$ are solvable?
44942
A group $G$ with a subgroup $H$ of index $n$ has a normal subgroup $K\subset H$ whose index in $G$ divides $n!$
45028
How to show that $\langle a \rangle$ is a subgroup of a group $G$, where $a$ is in center of the group $G$
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 $G=HK$?
106277
Prove that $G$ contains a subgroup $H_i$ for each $i\in\Bbb Z$ with $\dots\subset H_{−2}\subset H_{−1}\subset H_0\subset H_1\subset\dots?$
55350
H is a subgroup of G and G' is a subgroup of H. Prove H is normal in G.
121587
Let $H$ be a normal subgroup of index $n$ in a group $G$. Show that for all $g \in G, g^n \in H$
7575
Show the commutator subgroup of $S_{n}$ is $A_{n}$ for $n \geq 5$
35472
Transitive subgroup of symmetric group $S_n$ containing an $(n-1)$-cycle and a transposition
137691
What are the self-normalizing subgroups of $S_n$?
132342
Suppose $H$ is a subgroup of $S_n$ but $H$ is not a subgroup of $A_n$. Prove that |$H$ ∩ $A_n$| = $\frac{1}{2}|H|$?
144732
If $[G:H]=n$, then $g^{n!}\in H$ for all $g\in G$.
128834
Proving the number of even and odd permutations of a subgroup $H<S_{n}$ are equal, provided $H$ is not contained in $A_{n}$