Let $G\cong H$. If $\exists g\in G$ s.t. $|g|=n$, then $\exists h\in H$ with $|h|=n$, and if each element of $G$ is finite, then each in $H$ is too.
56234
Isomorphism $f$ preserves the order of an element?
43750
Subgroups of finite solvable groups. Solvable?
2761
If $S$ is a nonempty subset of group $G$, then $S^{|G|}$ is a subgroup of $G$.
23579
If $G$ is isomorphic to $H$, then ${\rm Aut}(G)$ is isomorphic to ${\rm Aut}(H)$
145511
Suppose $H$ is the only subgroup of order $o(H)$ in the finite group $G$. Prove that $H$ is a normal subgroup of $G$.
121587
Let $H$ be a normal subgroup of index $n$ in a group $G$. Show that for all $g \in G, g^n \in H$
63162
Is there a characterization of groups with the property $\forall N\unlhd G,\:\exists H\leq G\text{ s.t. }H\cong G/N$?
85129
If $H_1, H_2\leq G$ are such that $H_1\cong H_2$ then $G/H_1\cong G/H_2$?
27728
Question on a homomorphism of a set G.
95534
If $|G|=p^nq$, then $G$ contains a unique normal subgroup of index $q$
58186
Set theory proof, $\exists g(g\colon A\twoheadrightarrow C)$ or $\exists h(h\colon C\hookrightarrow B)$
58033
$G\times H\cong G$ with $H$ non-trivial
128834
Proving the number of even and odd permutations of a subgroup $H<S_{n}$ are equal, provided $H$ is not contained in $A_{n}$
19458
Finite groups and Normal subgroups - $G$ is a finite group such that $|G|=n$. Let $p$ be the smallest prime such that $p|n$.
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 $P_i\leq G_i$?
141043
$H$ normal in $G$. Need $G$ contain a subgroup isomorphic to $G/H$
14379
Let $H \leq G$. If $G = H H^x$ then $G=H$.