If N is a nilpotent subgroup of a group G and G' is a commutator of G then show that [N,G']=1
47179
Prove that $[N, G']=1$ when $N$ is a nilpotent subgroup of a group $G$?
3456
Commutator of two normal subgroups with trivial intersections
6115
Show that $GL_n(\mathbb{F})$ is a finite group iff $\mathbb{F}$ has a finite number of elements
129610
Why is the direct product of a finite number of nilpotent groups nilpotent?
76317
Let $C$ be the commutator subgroup of $G$. Prove that $G/C$ is abelian
98739
If $H$ is a subgroup of $G$ and $K$ is a subgroup of $H$, then $$|G:K|=|G:H||H:K|$$
119935
Commutator subgroup of a simple group - Fraleigh p. 152 15.19(h)
63882
Show that the heisenberg group is nilpotent
105170
About product of $k$ commutators
123610
How to show that the commutator subgroup is a normal subgroup
7324
Let $H$ be a subgroup of $G$ with $[G:H]=n$. Let $N$ be the kernel of the left-multiplication action on the cosets of $H$. Show $[G:N]$ divides $n!$
145159
Finite group is abelian if the representatives of its conjugacy classes commute
116125
A group $G$ with $G'$ abelian and every abelian normal subgroup finite
86336
For $G$ group and $H$ subgroup of finite index, prove that $N \subset H$ normal subgroup of $G$ of finite index exists
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 $G \cong H \times K$
138297
Let $G = GL_2\mathbb (R)$. Show that $T$ is a subgroup of $G$
30676
$G'$ and the product of all the elements in an odd order group