Number of conjugacy classes in generic finite group? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T12:07:54Z http://mathoverflow.net/feeds/question/34358 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/34358/number-of-conjugacy-classes-in-generic-finite-group Number of conjugacy classes in generic finite group? Jan Weidner 2010-08-03T07:48:07Z 2010-08-03T17:06:56Z <p>My knowlege in group theory is very limited and this question is out of pure curiousity: Given a random finite group, how many conjugacy classes will it probably have? I can try to make this question more precise:</p> <p>Define $a_n$ to be the average number of conjugacy classes in finite groups of order less or equal $n$.</p> <p>What is the asymptotic behaviour of the sequence $a_n$?</p> http://mathoverflow.net/questions/34358/number-of-conjugacy-classes-in-generic-finite-group/34413#34413 Answer by SandeepJ for Number of conjugacy classes in generic finite group? SandeepJ 2010-08-03T17:01:09Z 2010-08-03T17:06:56Z <p>In elementary terms, you have to analyze the following class equation $n = 1 + h_2 + ... + h_r$ where</p> <ul> <li>n is the order of the group G</li> <li>$h_k$ denotes the number of elements in the k-th conjugacy class, and $n = c_k.h_k$.</li> </ul> <p>Dividing by n, you get $1 = \frac{1}{n} + \frac{1}{c_2} + ... + \frac{1}{c_r}$ which has a finite number of solutions.</p> <p>Christine Ayoub in her paper <em>On the number of conjugate classes in a group</em> (Proc. Internat. Conf Theory of Groups Canberra 1967) has worked out this analysis for p-groups and there are probably more recent papers on this aspect, which Scott and others allude to in the comments. See for example</p> <ol> <li><em>MR2557143 Keller, Thomas Michael . Lower bounds for the number of conjugacy classes of finite groups. Math. Proc. Cambridge Philos. Soc. 147 (2009), no. 3, 567--577.</em></li> </ol> <p>Another way of looking at your question is to see that the number of conjugacy classes is the same as the number of irreducible representations. The <a href="http://en.wikipedia.org/wiki/Character_table" rel="nofollow">character table</a> is always square. Therefore, one could ask "<em>what are the number of irreducible characters Irr(G) in a finite group of order n?</em>". The number of linear characters are [G:G'] where G'=<a href="http://en.wikipedia.org/wiki/Commutator_subgroup" rel="nofollow">commutator subgroup</a> but the nonlinear ones are tougher and there are papers establishing various bounds for these.</p> <ol> <li><em>MR2526321 (2010d:20010) Aziziheris, Kamal ; Lewis, Mark L. Counting the number of nonlinear irreducible characters of a finite group. Comm. Algebra 37 (2009), no. 5, 1572--1578.</em></li> <li><em>MR0689258 (84d:20014) Wada, Tomoyuki . On the number of irreducible characters in a finite group. Hokkaido Math. J. 12 (1983), no. 1, 74--82.</em></li> <li><em>MR0798751 (87a:20006) Wada, Tomoyuki . On the number of irreducible characters in a finite group. II. Hokkaido Math. J. 14 (1985), no. 2, 149--154.</em></li> </ol>