Let $G$ be a group and let $H$ and $K$ be subgroups of $G$. The commutator subgroup $[H,K]$ is defined as the smallest subgroup containing all elements of the form $hkh^{−1}k^{−1}$, where $h \in H$ and $k \in K$. Pick out the true statement(s): 1. If $H$ and $K$ are normal subgroups, then $[H,K]$ is a normal subgroup. 2. If $H$ and $K$ are characteristic subgroups, then $[H,K]$ is a characteristic subgroup. 3. $[G,G]$ is normal in $G$ and $G/[G,G]$ is abelian.
Commutator subgroup problem
1
$\begingroup$
abstract-algebra
group-theory
-
2You can use $\TeX$ on this site by enclosing formulas in dollar signs; single dollar signs for inline formulas and double dollar signs for displayed equations. You can see the source code for any math formatting you see on this site by right-clicking on it and selecting "Show Math As:TeX Commands". [Here](http://meta.math.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference)'s a basic tutorial and quick reference. There's an "edit" link under the question. – 2012-09-16