Let $H$ and $K$ be normal subgroups of $G$, if $H \cap K = \{e\}$ prove that $hk = kh$ for all $h \in H, k \in K$
89659
$H_1 ,H_2 \unlhd \, G$ with $H_1 \cap H_2 = \{1_G\} $. Prove every two elements in $H_1, H_2$ commute
77361
How to prove $|H|[K:K \cap (g^{-1}Hg)]=|K|[H:H \cap (gKg^{-1})]$?
28108
Proving the kernel is a normal subgroup.
1220
Have $\alpha:G \to H$. If a subgroup $U$ is normal in $H$ prove the pre-image of $U$ is normal in $G$.
59566
Proving that $ka \equiv kb \pmod{m} \iff a \equiv b \pmod{m/\gcd(k,m)}$
105910
How to prove that $[G:xHx^{-1}] = [G:H]$ given $H \le G$?
102622
K is a subgroup of G. Show that K is not a normal subgroup
44934
Show that every normal subgroup can be written as an intersection of $\ker \chi$'s for $\chi \in \operatorname{Irr}(G).$
53475
Let $H$ have order $m$ and $K$ have order $n$, where $m$ and $n$ are relatively prime. Then $H \cap K=\{e\}$
100974
Showing H is a normal subgroup by calculating left and right coset
63780
to show characteristic subgroup is normal in G
106927
How do I prove $[G:H\cap K]\leq [G:H][G:K]$?
15276
$G \times H \cong G \times K$ , then $ K \cong H$
39165
Suppose $G$ is a group and $H,K \trianglelefteq G$. If $H \cap K =1$ the $hk=kh$ for all $k \in K, h \in H$?
147253
Prove for any $K \leq G$, either $K \leq N$ or $KN=G$ and $[K:K \cap N]=p$ where N is normal in G and $[G:N]=p$
12589
Show that every subgroup of the quaternion group Q is a normal subgroup of Q
131204
A non-trivial normal subgroup N of a finite group G or order $p^n$ is such that $N\cap Z(G)\neq \{ 1\}$