If $\gcd(|G:U|,|G:V|)=1$ for group $G$ and subgroup $U,V$ show $G=UV$.
91997
Product of two subgroups the whole group?
107935
If $G$ is a group, $H$ is a subgroup of $G$ and $g\in G$, is it possible that $gHg^{-1} \subset H$?
138297
Let $G = GL_2\mathbb (R)$. Show that $T$ is a subgroup of $G$
56245
Let H be a subgroup of a group G. Prove that the following statements are equivalent.
130112
If $\gcd(a,n) = 1$ and $\gcd(b,n) = 1$, then $\gcd(ab,n) = 1$.
136064
How can I prove that $\gcd(a,b)=1\implies \gcd(a^2,b^2)=1$ without using prime decomposition?
22845
Prove that if d is a common divisor of a and b, then $d=\gcd(a,b)$ if and only if $\gcd(a/d,b/d)=1$
76507
$\gcd(b^x - 1, b^y - 1, b^ z- 1,\dots) = b^{\gcd(x, y, z,\dots)} -1$
20679
Prove that if $g^2=e$ for all $g$ in $G$ then $G$ is Abelian.
150042
Prove $x^{-1}Hx$ is a subgroup of $G$
5789
$ d = \gcd(a,b)\Rightarrow\ \gcd(a/d,b/d) = 1$
13834
Show that G is a group, if G is finite, the operation is associative, and cancellation law holds
43728
Proof via Group Theory : $\mathrm{lcm}(a,b) \cdot \gcd(a,b) = |ab|$
134403
Let $G$ be a group, and let $n\in\mathbb Z$. Show that $(ab)^n=a^nb^n\iff(ab)^{1-n}=a^{1-n}b^{1-n}$.
57464
If $G$ is a group show that if $(a \cdot b)^2 = a^2 \cdot b^2$ then $G$ must be abelian.
150585
Group with $a=a^{-1}$ for all $a\in G$ is abelian
51858
Showing $\gcd(2^m-1,2^n+1)=1$