The smallest normal set containing a subset $X$, what is a "normal set"?
I've been self studying abstract algebra from these notes. I've ran across the following Lemma:
The confusing part is that $X$ is a "any subset" not any "subgroup", and that the term "normal set" appeared in place where I expected "normal subgroup". The definition of a "normal set" has not been provided and I couldn't find it with a quick Googling.
So the question is
$\endgroup$ 4What is a "normal set" and why is the proof of that Lemma obvious ?
1 Answer
$\begingroup$A normal set is a set $X$ for which $g^{-1}xg\in X$ for every $x\in X,g\in G$. It's just like the normality condition for subgroups, except that $X$ doesn't have to be a subgroup.
We can see that $\bigcup_{g\in G}g^{-1}Xg$ must be the smallest normal set containing $X$ quite easily.
Proceed algorithmically. Begin with $N=X$. We want $g^{-1}xg\in X$ for every $x\in X,g\in G$, so for each $x\in X$, add $g^{-1}xg$ to $N$. Repeat this process until all such elements are included in $N$. Then $N$ is the smallest normal set of $G$ containing $X$.
Of course, $\langle g^{-1}Xg : g \in G \rangle$ would be the smallest normal subgroup of $G$ containing $X$.
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