# Normal subgroup notation

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.

• In this case you should denote the inverse by $(-x)$, not $x^{-1}$. – Mark Jan 17 at 15:17
• 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 . – An Isomorphic Teen Jan 17 at 15:19
• 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. – Mark Jan 17 at 15:20

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 $$-x$$.
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.