Prove that if $H$ is a subgroup of group $G$, $H\circ x=H$ if and only if $x\in H$
93666
Does the closedness of a subgroup $H$ of $G$ imply that for $h\in H, g\in G \setminus H : hg \not \in H$?
144065
Proving $\phi(G)$ is cyclic if $\phi$ : $G \to H$ is group homomorphism and $G$ is cyclic
145511
Suppose $H$ is the only subgroup of order $o(H)$ in the finite group $G$. Prove that $H$ is a normal subgroup of $G$.
154878
Prove that $H$ is a normal subgroup in the finite group $G=\langle S\rangle $ iff $sHs^{-1}$ $\subseteq H$ for all $s\in S$.
103002
Group equals union of two subgroups
2761
If $S$ is a nonempty subset of group $G$, then $S^{|G|}$ is a subgroup of $G$.
80273
Prove the centralizer of an element in group $G$ is a subgroup of $G$
33387
Proof that $n\Bbb Z \leq \Bbb Z$ and are the only subgroups of $\Bbb Z$
86336
For $G$ group and $H$ subgroup of finite index, prove that $N \subset H$ normal subgroup of $G$ of finite index exists
65037
$H$ is a maximal normal subgroup of $G$ if and only if $G/H$ is simple.
59561
Suppose that G is a group and N < G a normal subgroup. Prove G/N is simple
41237
$H$ is normal subgroup of $G$ iff $gHg^{-1}=H$ for every $g$ of $G$
150042
Prove $x^{-1}Hx$ is a subgroup of $G$
29996
Prove that if a normal subgroup $H$ of $ G$ has index $n$, then $g^n \in H$ for all $g \in G$
102859
Proving that $G$ is a group if $a*x=b$ and $y*a=b$ have solutions.
52105
Prove that $HK$ is a subgroup iff $HK=KH$.
122581
Prove or disprove that $G_1/H_1 \cong G_2/H_2$