Assume $(G,\times)$ is a group and for $a,b \in G$: $ab=ba$, $\text{ord}(a)=n$, $\text{ord} (b)=m$
74187
$\operatorname{ord}(u)=r,\,\operatorname{ord}(v)=s$ $\,\Rightarrow\,\operatorname{ord}(uv)=rs\,$ if $\,r,s\,$ (co)primes
78678
If $\gcd(m,n)=1$, and $\gcd(a,m)=\gcd(b,n)=1$, then $\gcd(ab, mn)=1$
157040
For groups $A,B,C$, if $A\times B$ and $A\times C$ are isomorphic do we have $B$ isomorphic to $C$?
150585
Group with $a=a^{-1}$ for all $a\in G$ is abelian
134403
Let $G$ be a group, and let $n\in\mathbb Z$. Show that $(ab)^n=a^nb^n\iff(ab)^{1-n}=a^{1-n}b^{1-n}$.
105180
When does $(ab)^n = a^n b^n$ imply a group is abelian?
24181
Showing that $G$ is a group under an alternative operation.
2235
Showing $\text{Aut}_\mathsf{Grp}(\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}) \cong S_3$
58033
$G\times H\cong G$ with $H$ non-trivial
15276
$G \times H \cong G \times K$ , then $ K \cong H$
25713
If $a$ and $b$ commute and $\text{gcd}\left(\text{ord}(a),\text{ord}(b)\right)=1$, then $\text{ord}(ab)=\text{ord}(a)\text{ord}(b)$.
128652
Subgroups of $\mathbb Z \times(\mathbb Z/n\mathbb Z)$
26861
$a+b=c \times d$ and $a\times b = c + d$
139833
Prove that $\gcd(M, N)\times \mbox{lcm}(M, N) = M \times N$.
30676
$G'$ and the product of all the elements in an odd order group
67772
How to find $k$ such that $(\mathbb{Z} \times \mathbb{Z})/ \langle (m,n)\rangle \cong \mathbb{Z} \times \mathbb{Z}_k$
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 $G \cong H \times K$