Exercise 4.3.27 (If the representatives of the conjugacy classes of $G$ pairwise commute, then $G$ is abelian)

Question

Let $g_1,g_2,\ldots,g_r$ be representatives of the conjugacy classes of the finite group $G$ and assume these elements pairwise commute. Prove that $G$ is abelian.

Richard Ganaye · Accepted Answer

\begin{proof} 
Consider the subgroup $H = \langle g_1, g_2, \ldots, g_r\rangle$.

Since every element  $ g \in G$ is of the form $g = a g_i a^{-1}$ for some $a \in G$ and some $i$, $1\leq i \leq r$, we obtain
$$\bigcup_{g\in G} gHg^{-1} = G.$$
Then Exercise 24 shows that $H$ is not a proper subgroup of $G$, thus $H = G$, so
$$G = \langle g_1, g_2, \ldots, g_r\rangle.$$
Since $g_i g_j = g_j g_i$ for all indices $i,j$, this shows that $G$ is abelian.
\end{proof}

Exercise 4.3.27 (If the representatives of the conjugacy classes of $G$ pairwise commute, then $G$ is abelian)

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Exercise 4.3.27 (If the representatives of the conjugacy classes of $G$ pairwise commute, then $G$ is abelian)

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