$H$ is a subgroup of $G$, $H\neq G$ then $G\neq\bigcup_{g\in G} H^g$
72150
Union of the conjugates of a proper subgroup
15276
$G \times H \cong G \times K$ , then $ K \cong H$
29996
Prove that if a normal subgroup $H$ of $ G$ has index $n$, then $g^n \in H$ for all $g \in G$
42891
Prove that $Z(G)$ which is the center of $G$ is a subgroup of $G$
94571
How can we prove that $N_G(H)=H$ implies $N_{f(G)}(f(H))=f(H)$
130455
If $H,K⊲G$ and $H∩K = \{1_G\}$, then all elements in $H$ commute with all elements in $K$
40094
Does $G\oplus G \cong H\oplus H$ imply $G\cong H$ in general?
87882
If $N \lhd G$ is abelian and $G/N$ is abelian then $G$ is abelian
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 $G$.
58033
$G\times H\cong G$ with $H$ non-trivial
95534
If $|G|=p^nq$, then $G$ contains a unique normal subgroup of index $q$
99630
For every $g\in G$ there exists an $h\in G$ such that $g = h^3$
140334
Let $G$ be a group and $H$ be a subgroup of $G$ of index $2$ .Then show that if $a,b\notin H$ then $ab\in H$.
98816
How to prove that $\{f+g<a\}=\bigcup_{r\in{\Bbb Q}}(\{f<r\}\cap\{g<a-r\})$?
67199
Isomorphism $H_n(D^n, S^{n - 1};A)$ and $A$
111703
If $H \leq G$ and $[G:H]! \leq |G|$ then $G$ is not simple
56245
Let H be a subgroup of a group G. Prove that the following statements are equivalent.