If $H,K⊲G$ and $H∩K = \{1_G\}$, then all elements in $H$ commute with all elements in $K$
23941
Normal Subgroups that Intersect Trivially
105951
Does $G\times K\cong H\times K$ imply $G\cong H$?
39165
Suppose $G$ is a group and $H,K \trianglelefteq G$. If $H \cap K =1$ the $hk=kh$ for all $k \in K, h \in H$?
151621
$G$ a group and $H, K\mathrel{\unlhd}G$. Assuming that $H \cap K = \{1_G\}$ and $G = \langle H, K \rangle$, prove that $G \cong H \times K$
74460
Matrices A+B=AB implies A commutes with B
41237
$H$ is normal subgroup of $G$ iff $gHg^{-1}=H$ for every $g$ of $G$
103106
Proving that any permutation in $S_n$ can be written as a product of disjoint cycles
29996
Prove that if a normal subgroup $H$ of $ G$ has index $n$, then $g^n \in H$ for all $g \in G$
25713
If $a$ and $b$ commute and $\text{gcd}\left(\text{ord}(a),\text{ord}(b)\right)=1$, then $\text{ord}(ab)=\text{ord}(a)\text{ord}(b)$.
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 $G$.
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 $[G:N]=p$
158355
If $A$ commutes with $B$ and $B$ commutes with $C$, then $A$ commutes with $C$
91777
Product of Non-Disjoint Cycles
94507
Given $G$ group and $H \le G$ such that $|G:H|=2$, how does $x^2 \in H$ for every $x \in G$?
144732
If $[G:H]=n$, then $g^{n!}\in H$ for all $g\in G$.
73174
Prove $KN\cong K\times N$ if $K\cap N=\{e\}$
94571
How can we prove that $N_G(H)=H$ implies $N_{f(G)}(f(H))=f(H)$