If $G$ is a group, $H \leq G$ and $K \leq G$, then $HK \leq G \iff HK = KH$.
52105
Prove that $HK$ is a subgroup iff $HK=KH$.
80186
How to show that $K\leq S_4$ is a normal subgroup?
79826
$gHg^{-1}\subset H$ whenever $Ha\not = Hb$ implies $aH\not =bH$
102979
If $[G:H]$ and $[G:K]$ are relatively prime, then $G=HK$
106277
Prove that $G$ contains a subgroup $H_i$ for each $i\in\Bbb Z$ with $\dots\subset H_{−2}\subset H_{−1}\subset H_0\subset H_1\subset\dots?$
94571
How can we prove that $N_G(H)=H$ implies $N_{f(G)}(f(H))=f(H)$
65744
How can I Prove that H=G or K=G
141043
$H$ normal in $G$. Need $G$ contain a subgroup isomorphic to $G/H$
34724
For $G$ a group and $H\unlhd G$, then $G$ is solvable iff $H$ and $G/H$ are solvable?
124709
Suppose G be a non-abelian group and H,K be two abelian subgroups of G. Then must HK be an abelian subgroup of G?
124309
$g\in G$ maximal order in $G$ abelian then $G=\left\oplus H$
7121
If $G$ is a finite group and $|G| < |A| + |B|$, then $G=AB$.
79933
Let $G$ be a group and $c\in G$. Define $*$ on $G$ by: $x*y=xdy$ for all $x,y\in G$, where $d$ is the inverse of $c$. Prove $(G,*)$ is a group.
15276
$G \times H \cong G \times K$ , then $ K \cong H$
150199
How to prove for a finite group that $|\langle g \rangle|=o(g)$?
129452
Groups $G,H$ with surjective homomorphisms that are quotient of each other
77221
Prove that $G$ has a subgroup isomorphic to $G/H$.