$H\trianglelefteq A$ and $K\trianglelefteq B\Rightarrow HK\trianglelefteq AB$?

Question

Let $G$ be a group and $A, B\leq G$. Suppose $H\trianglelefteq A$ and $K\trianglelefteq B$. Is it true that $HK\trianglelefteq AB$?

Notation: $\leq$ means subgroup and $\trianglelefteq$ means normal subgroup.

Thanks

$A$ and $B$ are subgroups of some larger group $G$? – Exodd Oct 19 '14 at 17:09 — Exodd, Oct 19 '14 at 17:09

Asinomás · Answer 1 · 2014-10-19T17:17:19.307

4

It is false. Take $H=\{1\}, A=G,$ and $K=B$. If the assumption was true we would have $K\lhd G$ but $K$ can be any subgroup.

edited Oct 19 '14 at 17:17

answered Oct 19 '14 at 17:11

Asinomás

105,651

1

where is your $g$? Don't forget the normality of $H$ and $K$ is with respect to $A$ and $B$, respectively.. – PtF Oct 19 '14 at 17:12
In other words, if this was true all subgroups would be norma – Asinomás Oct 19 '14 at 17:18
Very nice indeed. +1 – Timbuc Oct 19 '14 at 17:20
Yes, I agree... – PtF Oct 19 '14 at 17:20

score 1 · Answer 2 · answered Oct 19 '14 at 17:17

1

No.

First of all, the set $AB$ is not in general a subgroup, consider eg $H = A$ and $K = B$ for easy examples.

Even if it is, the claim is false. Consider $G = B$, $H = A$. The claim fails since it is possible to find examples where $H$ and $K$ are subgroups of $G$, $K$ is normal in $G$, but $HK$ is not a normal subgroup.

answered Oct 19 '14 at 17:17

spin

11,956

You're right $HK$ is a subgroup of $g$ if and only if $HK=KH$.. – PtF Oct 19 '14 at 17:20

$H\trianglelefteq A$ and $K\trianglelefteq B\Rightarrow HK\trianglelefteq AB$?

2 Answers2