Commuting Conjugacy Classes' Representatives (4.3.27)

Dummit and Foote Abstract Algebra, section 4.3, exercise 27:

Let g₁, g₂, ..., g_r be the representatives of the conjugacy classes of the finite group G and assume that these elements commute pairwise. Prove that G is abelian.

Proof: Since there are r conjugacy classes in G, we have that each conjugacy class is, on average, of size | G | / r. Since we have that r ≤ | C_G(g_i) | for all i ≤ r, we have that each conjugacy class is at the most of size | G : C_G(g) | ≤ | G | / r. In order to fulfill the average requirement, every group must be of the average size, and consequently of the same size. Since the conjugacy class of the identity element is of size 1, that implies that every conjugacy class is of size 1, i.e. C_G(g) = G for all g∈G, therefore G is abelian.

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