# Questions tagged [derived-subgroup]

Also called the commutator subgroup of a group is the subgroup generated by all commutator elements of that group. Should be used with the (group-theory) tag.

56 questions
Filter by
Sorted by
Tagged with
22 views

### Closure of the commutator group of Lie solvable group

Help me please to understand the following. Let $G$ be a connected Lie solvable group. Let $G':=\overline{[G,G]}$ be a closer of the commutator group $[G,G]$. Is it possible that $G'=G$? If so what if ...
56 views

### Can we write a subgroup as a product of subgroups? [duplicate]

Let $G$ be a group such that $$G=\prod G_i,$$ where above product is arbitrary. Suppose that there exists a subgroup $H$ of $G$ such that $$\prod G_i=H\prod G_i',$$ where $G_i'$ is derived subgroup of ...
54 views

### Is there an easy way to find the derived group of $S_5$?

In order to find the derived group of $S_5$ I've tried using Lagrange’s Theorem to find the order of the possible subgroups but $O(S_5)=2^3\cdot 3 \cdot 5$ so there are too many possible subgroups to ...
### How to find the derived subgroup of $Q_8$
Given the group $Q_8=\{ \pm1, \pm i, \pm j, \pm k \}$ is there an easy way to find its derived subgroup? I've tried by hand and this seems such a big task to do manually. After this, by thinking a ...