Let $G$ be a finite group. Let $X_n$ be the set $$X_n = \{(g_1, g_2, \dots, g_n)\in G^n : g_i g_j = g_j g_i, 1 \leq i, j \leq n\}.$$
- Show that $|G|$ divides $|X_n|.$
- Compute $X_2$ when $G$ is $S_5.$
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Ricci Ten
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You can speed up the calculation by restricting first to tuples in which $g_1$ is a representative of a conjugacy class for $S_5$. – Derek Holt Jan 13 '25 at 16:22
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i want to solve part b of this question – Ricci Ten Jan 13 '25 at 16:30
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@DerekHolt can you elaborate – Ricci Ten Jan 13 '25 at 16:32
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Hint (?): $X_2=\cup_{g\in G}({g}\times C(g))$ where $C(g)$ denotes the centralizer of $g$ in $G$, i.e. $C(g)={h\in G\mid gh=hg}.$ – Anne Bauval Jan 13 '25 at 16:38
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@DerekHolt OP is asked to compute $X_2$ when $G=S_5$, not just $|X_2|$. He says he wants to do this "explicitely", but this doesn't seem realistic ($X_2$ is too huge). Imo he should just describe $C(g)$ for each kind of cycle decomposition of $g$. – Anne Bauval Jan 13 '25 at 16:50
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Sorry I meant $|X_2|/|G|$ is equal to the number of conjugacy classes of $G$. This is a comment, not an answer to the question. – Derek Holt Jan 13 '25 at 16:56
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After my hint, this question could be considered as a duplicate of Centralizer of a permutation in $ Sym(n) $.. – Anne Bauval Jan 13 '25 at 17:03