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1987•Proceedings of the Royal Society of Edinburgh Section A MathematicsRequires access

Conjugacy classes in finite groups

Antonio Vera López

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Abstract

Synopsis In this paper, the number of conjugacy classes in a finite group G is analysed in terms of the number of ordered pairs that generate it. Using this relation, we give a new elementary proof of one of A. Mann's results for finite groups, namely: |G| ≡ r(G) (mod. d|G|. δ|G|), where , prime and pi ≠ Pj for every i≠j, r(G) denotes the number of conjugacy classes of elements of G, d|G| = g.c.d. (p1 − 1, … pt − 1) and δ|G| = g.c.d. . The above congruence is obtained without using character theory. We also obtain new local congruences that slightly improve Mann's congruence.

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Synopsis In this paper, the number of conjugacy classes in a finite group G is analysed in terms of the number of ordered pairs that generate it. Using this relation, we give a new elementary proof of one of A. Mann's results for finite groups, namely: |G| ≡ r(G) (mod. d|G|. δ|G|), where , prime and pi ≠ Pj for every i≠j, r(G) denotes the number of conjugacy classes of elements of G, d|G| = g.c.d. (p1 − 1, … pt − 1) and δ|G| = g.c.d. . The above congruence is obtained without using character theory. We also obtain new local congruences that slightly improve Mann's congruence.

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Available abstract

Synopsis In this paper, the number of conjugacy classes in a finite group G is analysed in terms of the number of ordered pairs that generate it. Using this relation, we give a new elementary proof of one of A. Mann's results for finite groups, namely: |G| ≡ r(G) (mod. d|G|. δ|G|), where , prime and pi ≠ Pj for every i≠j, r(G) denotes the number of conjugacy classes of elements of G, d|G| = g.c.d. (p1 − 1, … pt − 1) and δ|G| = g.c.d. . The above congruence is obtained without using character theory. We also obtain new local congruences that slightly improve Mann's congruence.

Key concepts: Conjugacy class, Congruence relation, Mathematics, Finite group, Combinatorics, Congruence (geometry), Character table, Prime (order theory)

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