On conjugacy classes of groups of Squarefree order

TL;DR

This paper derives explicit formulas and bounds for the class number of groups of squarefree order, enabling efficient computation of the maximum size of such groups for given class numbers, with practical results up to class number 100.

Contribution

It provides a new explicit formula for the class number of squarefree order groups and an improved upper bound on their size, along with an algorithm and implementation for computation.

Findings

Derived an explicit formula for class number of squarefree order groups.

Established an upper bound |G| ≤ k(G)^3 for these groups.

Computed maximal squarefree group sizes for class numbers up to 100.

Abstract

The problem of finding the largest finite group with a certain class number (number of conjugacy classes), $k(G)$, has been investigated by a number of researchers since the early 1900's and has been solved by computer for $k(G) \leq 9$. (For the restriction to simple groups for $k(G) \leq 12$.) One has also tried to find a general upper bound on $|G|$ in terms of $k(G)$. The best known upper bound in the general case is in the order of magnitude $|G| \leq k(G)^{2^{k(G)-1}}$. In this paper we consider the restriction of this longstanding problem to groups of squarefree order. We derive an explicit formula for the class number $k(G)$ of any group of squarefree order and we also obtain an estimate $|G| \leq k(G)^3$ in this case. Combining the two results we get an efficient way to compute the largest squarefree order a group with a certain class number can have. We also provide an implementation of this algorithm to compute the maximal squarefree $|G|$ for arbitrary $k(G)$ and display the results obtained for $k(G) \leq 100$.