$H < G $, if H is normal, then $H=gHg^{-1}$
41237
$H$ is normal subgroup of $G$ iff $gHg^{-1}=H$ for every $g$ of $G$
59561
Suppose that G is a group and N < G a normal subgroup. Prove G/N is simple
14379
Let $H \leq G$. If $G = H H^x$ then $G=H$.
94507
Given $G$ group and $H \le G$ such that $|G:H|=2$, how does $x^2 \in H$ for every $x \in G$?
89659
$H_1 ,H_2 \unlhd \, G$ with $H_1 \cap H_2 = \{1_G\} $. Prove every two elements in $H_1, H_2$ commute
45310
If $H\leq Z (G)$ and $G/H$ is nilpotent, then $G$ is nilpotent.
144732
If $[G:H]=n$, then $g^{n!}\in H$ for all $g\in G$.
35800
If $K \leq H \leq G$, show that $[G:K] = [G:H][H:K]$.
34567
If $G_1$ and $G_2$ are finite group and $H\leq G_1\times G_2$ then $H=P_1\times P_2$ with $P_i\leq G_i$?
28722
Prove that $H$ is a abelian subgroup of odd order
122581
Prove or disprove that $G_1/H_1 \cong G_2/H_2$
110672
Is it true that $aH = bH$ iff $ab^{-1} \in H$
75254
Properties possessed by $H , G/H$ but not G
23579
If $G$ is isomorphic to $H$, then ${\rm Aut}(G)$ is isomorphic to ${\rm Aut}(H)$
25404
Let $G$ be a group of order $2m$ where $m$ is odd. Prove that $G$ contains a normal subgroup of order $m$
105910
How to prove that $[G:xHx^{-1}] = [G:H]$ given $H \le G$?
44152
If $G$ has only one subgroup of order $n$, then that Subgroup is Normal