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I came upon this answer by pjs36 while studying normal subgroups, where they state the following theorem:

Let $G$ be a group, and $H≤G$ a subgroup of G. Then $H$ is normal in $G$ if and only if $H$ is a union of conjugacy classes of $G$.

Afterwards the following explanation is offered:

The above theorem about normal subgroups being a union of conjugacy classes is probably the most elementary way to prove that the alternating group A5 is simple. Its conjugacy class sizes are $1,12,12,15$, and $20$: There are no ways to add these numbers up (including the $1$, for the identity!) and get a divisor of $60$, hence no nontrivial, proper, normal subgroups.

And I began to wonder, for a general finite group $ G $ what are the necessary conditions for a union of its conjugacy classes to become a normal subgroup?

More specifically, let $G$ be a group such that $|G|=n$ and let $\operatorname{Cl}(a)=\{gag^{-1} : g\in G\}$ be the conjugacy class of some $a\in G$.

Then for a set of different class representatives of some size $R=\{e,g_1,g_2,...,g_{r-1}\}\subset G$ where $ 1\leq r\leq n$ let $A=\bigcup_{g_i \in R}\operatorname{Cl}(g_i)$ be the union of their conjugacy classes.

Is it enough for ${\vert A\vert } $ to divide $n$ for it to be a subgroup of $G$, and thus a normal one? Or do we need some other conditions?

Bernard
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Bamapag
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  • Related. – Shaun Sep 11 '20 at 13:45
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    It is certainly not enough for $|A|$ to divide $n$. For example, if that were the case, it would imply that any subset of a finite abelian group $G$ that contains the identity and has order dividing $|G|$ is a subgroup, which is definitely not the case. – Arturo Magidin Sep 11 '20 at 17:19
  • @ArturoMagidin That makes sense, for example $\mathbb{Z}_{12}^\times$ is abelian but ${1,7,11}$ is not a subgroup since it lacks $7\cdot 11=5\operatorname{mod}12$, while it is a union of conjugacy class since each element is its own class. Is it possible to find additional conditions that are weaker than outright demanding the Group's definitions to hold? – Bamapag Sep 12 '20 at 09:44

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