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According to the splitting criterion for conjugacy classes in the alternating group, a conjugacy class in $S_n$ of even permutations (i.e., a conjugacy class that lies completely inside $A_n$) splits into two conjugacy classes in $A_n$ if a certain condition (expressible in two different languages - centralizers and cycle decompositions) does hold. May we expect that when the splitting takes place, the two classes have the same size$^\dagger$? Or are there counterexamples?

$^\dagger$For example, this is true for the conjugacy class of $S_4$ made of the 8 $3$-cycles, which splits as $4+4$ in $A_4$.

Kan't
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1 Answers1

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Here is the general observation.

Let $G$ be a group and let $N$ be a normal subgroup. If $G$ acts on transitively a set $X$, then $G$ acts transitively on the $N$-orbits on $X$.

For simplicity of notation suppose that $G$ is finite (you can easily generalize this to infinite groups). Denote the $G$-conjugacy class of $g \in G$ by $g^G$.

Then for $g \in G$ the conjugacy class splits as $$g^G = g_1^N \cup \cdots \cup g_t^N$$ where $g_1^N$, $\ldots$, $g_t^N$ are the $N$-conjugacy classes in $g^G$.

Here $G$ acts transitively (by conjugation) on $g^G$, so by the general observation mentioned earlier it acts transitively on $\{g_1^N, \ldots, g_t^N\}$. In particular $|g_i^N| = |g_j^N|$ for all $i,j$.

testaccount
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