3
$\begingroup$

Let $G$ be a group and $H$ a subgroup, then it is standard to say that $H$ is normal in $G$ if and only if for all $x \in G$, $xHx^{-1} \subseteq H$. My question here is, if $G$ and $H$ are additive groups, it is okay notation to write $x + H + x^{-1}$? Or is this something that is not done.

$\endgroup$
3
  • 2
    $\begingroup$ In this case you should denote the inverse by $(-x)$, not $x^{-1}$. $\endgroup$ – Mark Jan 17 at 15:17
  • $\begingroup$ Oh yes indeed. But is the notation acceptable? Or is it better off to use the multiplication notation even though i am referring to additive groups . $\endgroup$ – An Isomorphic Teen Jan 17 at 15:19
  • 6
    $\begingroup$ If the group is additive then we indeed write $+$. But remember that when we call the operation addition, we usually mean the group is abelian, and in this case every subgroup is normal. $\endgroup$ – Mark Jan 17 at 15:20
2
$\begingroup$

For any binary operation $\ast$ that defines a group $G$ with a subgroup (or even a subset) $H$, it is acceptable to write

$$x\ast H\ast x^{-1}=\{ x\ast h\ast x^{-1}\mid h\in H\}$$

for any $x\in G$. This is due to the associativity of $\ast$. Note that $x^{-1}$ is with respect to $\ast$, so if $\ast=+$, then use $-x$.

$\endgroup$
3
$\begingroup$

As Mark points out in the comments, it would be more appropriate to write $x+H+(-x)$ in such a case, but we typically use the additive notation for abelian groups, in which case every subgroup is normal, so we wouldn't bother to write it.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.