If $\phi:G\to H$ is an isomorphism, prove that $o(x)=o(\phi(x))$ for all $x\in G$
56234
Isomorphism $f$ preserves the order of an element?
140962
Counterexamples for $G \times H$ isomorphic to $G$ implies $H$ trivial.
5042
Order of a product of subgroups. Prove that $o(HK) = \frac{o(H)o(K)}{o(H \cap K)}$.
23579
If $G$ is isomorphic to $H$, then ${\rm Aut}(G)$ is isomorphic to ${\rm Aut}(H)$
152887
Let $T : V \rightarrow V$ be a linear operator. If $T^n = O_V$ for some $n \ge 1$, prove that $I_V + T$ is an isomorphism.
77361
How to prove $|H|[K:K \cap (g^{-1}Hg)]=|K|[H:H \cap (gKg^{-1})]$?
77221
Prove that $G$ has a subgroup isomorphic to $G/H$.
61513
Prove that all cyclic groups of the same order are isomorphic
151047
a non trivial example: $G/H$ is isomorphic to $G$
145511
Suppose $H$ is the only subgroup of order $o(H)$ in the finite group $G$. Prove that $H$ is a normal subgroup of $G$.
100439
Proving that a Galois group $Gal(E/Q)$ is isomorphic to $\mathbb{F}_p^\times$
141043
$H$ normal in $G$. Need $G$ contain a subgroup isomorphic to $G/H$
52556
Isomorphism of $S^1$.
58663
How to show that if $\phi(n)$ equals to n itself, then n must be 1?
154398
Show that H is a subgroup of G
152603
Group isomorphic to its automorphism group
86823
proof verification: If $f:G \rightarrow H$ is group homomorphism, and $H$ is abelian, then $G$ is abelian