Example of subgroup $H$ with the property $g H g^{-1}$ is properly contained in $H$.
41085
Conjugate subgroup strictly contained in the initial subgroup?
79826
$gHg^{-1}\subset H$ whenever $Ha\not = Hb$ implies $aH\not =bH$
23579
If $G$ is isomorphic to $H$, then ${\rm Aut}(G)$ is isomorphic to ${\rm Aut}(H)$
55976
Are groups satisfying $g=g^{-1}$ for all $g \in G$ abelian?
156463
The complement of a proper subgroup generates the whole group
106927
How do I prove $[G:H\cap K]\leq [G:H][G:K]$?
63780
to show characteristic subgroup is normal in G
132342
Suppose $H$ is a subgroup of $S_n$ but $H$ is not a subgroup of $A_n$. Prove that |$H$ ∩ $A_n$| = $\frac{1}{2}|H|$?
151621
$G$ a group and $H, K\mathrel{\unlhd}G$. Assuming that $H \cap K = \{1_G\}$ and $G = \langle H, K \rangle$, prove that $G \cong H \times K$
95867
Subgroup of $C^*$ (nonzero complex) with finite index.
90616
When is mapping $g$ to $g^{-1}$ a group homomorphism?
129120
Example of a group $G$ such that $G = N_1\cdots N_n$ and $N_i \cap N_j = \{e\}$ for all $i \neq j$ but $G$ is not the internal direct product of them.
85129
If $H_1, H_2\leq G$ are such that $H_1\cong H_2$ then $G/H_1\cong G/H_2$?
27436
Does $G\cong G/H$ imply that $H$ is trivial?
44942
A group $G$ with a subgroup $H$ of index $n$ has a normal subgroup $K\subset H$ whose index in $G$ divides $n!$
34902
If $H$ is a cyclic subgroup of $G$ and $H$ is normal in $G$, then every subgoup of $H$ is normal in $G$.
72150
Union of the conjugates of a proper subgroup