If $|G|< \infty$ and $N\lhd G$ such that $G/N$ is nilpotent, then there exists a nilpotent subgroup $H\leq G$ such that $G=HN$.
61526
An exercise about nilpotent quotient groups
59561
Suppose that G is a group and N < G a normal subgroup. Prove G/N is simple
22477
If H is a subgroup of a group G and K is a normal subgroup of G the show that K is a normal subgroup of HK.
85011
Prove that $\exists a,g,h\in G\colon h=aga^{-1}, g\neq h ,gh=hg$ in a finite non-abelian group $G$.
106779
How to show if $H,K$ are normal subgroups in a group $G$ then $[H,K] \leq H \cap K$
133001
$G$ simple group and there exists a subgroup of index $n$. Show that: $|G|$ divides $n!/2$
144218
Does every finite nilpotent group occur as a Frattini subgroup?
135808
Show that there is a positive integer $n$ such that $G \cong \ker(\phi^{n}) \times \phi^{n}(G)$
91774
Existence of group $G$ with $|G|>2$ and automorphism group of odd order
42389
If $(G, \oplus)$ has order $2n$, prove that every proper subgroup of $(G, \oplus)$
68317
Conjugacy class of $G$ splitting as Conjugacy class of $H\unlhd G$
87882
If $N \lhd G$ is abelian and $G/N$ is abelian then $G$ is abelian
23579
If $G$ is isomorphic to $H$, then ${\rm Aut}(G)$ is isomorphic to ${\rm Aut}(H)$
61379
Let $G$ be abelian, $H$ and $K$ subgroups of orders $n$, $m$. Then G has subgroup of order $\operatorname{lcm}(n,m)$.
151130
When is the product of $n$ subgroups a subgroup?
114221
A Group $G$ is soluble $\iff G^{(k)}=\{e\}$ for some $k$
29996
Prove that if a normal subgroup $H$ of $ G$ has index $n$, then $g^n \in H$ for all $g \in G$