$\frac{HS}{H\cap S}$ is a Sylow $p-$subgroup of $\frac{G}{H}$.
117744
$p$-Sylow in quotient groups
91287
Is $G/pG$ is a $p$-group?
154878
Prove that $H$ is a normal subgroup in the finite group $G=\langle S\rangle $ iff $sHs^{-1}$ $\subseteq H$ for all $s\in S$.
11825
Let $H$ be a subgroup of $G$. Let $K = \{x \in G: xax^{-1} \in H \iff a \in H\}$. Prove that $K$ is a subgroup of $G$.
19720
A normal subgroup intersects the center of the $p$-group nontrivially
94469
How to prove that $|HK| = \dfrac{|H| \; |K|}{|H \cap K|}$?
145511
Suppose $H$ is the only subgroup of order $o(H)$ in the finite group $G$. Prove that $H$ is a normal subgroup of $G$.
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 $G$
64269
Finding the minimal $n$ such that a given finite group $G$ is a subgroup of $S_n$
106277
Prove that $G$ contains a subgroup $H_i$ for each $i\in\Bbb Z$ with $\dots\subset H_{−2}\subset H_{−1}\subset H_0\subset H_1\subset\dots?$
106779
How to show if $H,K$ are normal subgroups in a group $G$ then $[H,K] \leq H \cap K$
27215
Intersection of the $p$-sylow and $q$-sylow subgroups of group $G$
99694
Number of subgroups of $S_4$
35353
Isomorphic quotient groups $\frac{G}{H} \cong \frac{G}{K}$ imply $H \cong K$?
11731
Normal subgroups of $S_4$
151047
a non trivial example: $G/H$ is isomorphic to $G$
151727
If $|G/H|=4$ then $G$ is union of three proper subgroups