Prove that there is a subgroup $K$ : $k! | [H:K]$
44942
A group $G$ with a subgroup $H$ of index $n$ has a normal subgroup $K\subset H$ whose index in $G$ divides $n!$
39469
If $P \leq G$, $Q\leq G$, are $P\cap Q$ and $P\cup Q$ subgroups of $G$?
35800
If $K \leq H \leq G$, show that $[G:K] = [G:H][H:K]$.
106927
How do I prove $[G:H\cap K]\leq [G:H][G:K]$?
10602
Monotone Union of subgroups being subgroup
131516
Subgroup generated by n-th powers of elements
106277
Prove that $G$ contains a subgroup $H_i$ for each $i\in\Bbb Z$ with $\dots\subset H_{−2}\subset H_{−1}\subset H_0\subset H_1\subset\dots?$
114431
Monoids and Subgroups
1251
If $[G:K]$ is finite, then $[H:H \cap K] = [G:K]$ iff $G = HK$ (Hungerford Proposition 4.8, Proof)
141683
$ |G_1 |$ and $|G_2 | $ are coprime. Show that $K = H_1 \times H_2$
64408
Subgroup of $\mathbb{Q}$ with finite index
106779
How to show if $H,K$ are normal subgroups in a group $G$ then $[H,K] \leq H \cap K$
85129
If $H_1, H_2\leq G$ are such that $H_1\cong H_2$ then $G/H_1\cong G/H_2$?
154878
Prove that $H$ is a normal subgroup in the finite group $G=\langle S\rangle $ iff $sHs^{-1}$ $\subseteq H$ for all $s\in S$.
41237
$H$ is normal subgroup of $G$ iff $gHg^{-1}=H$ for every $g$ of $G$
23533
All subgroups normal $\implies$ abelian group
42389
If $(G, \oplus)$ has order $2n$, prove that every proper subgroup of $(G, \oplus)$