What are bounds on number of conjugacy classes in terms of number of elements of a group ?
(I allowed myself to edit the question in spirit of remarkable answers given to it by Gerry Myerson and Geoff Robinson. Below is original text of the question. (Alexander Chervov) ).
It's about the first step to find an upper bound to the order of a finite group with h conjugacy classes (right or left) that depends only on h. (h a natural non nul integer).
I have some doubts about the rigor of my proof that I am sharing with you so that you can help me find a likely error or an omited step.
I have attached the scan of my proof to this post.
Many thanks
Best Answer
There is a theorem of E. Landau which proves that if you fix a positive integer h, there are only finitely many finite groups with h conjugacy classes. This proof is more number theory than group theory, in fact. More recently, one person who has worked more extensively on this question using more group theory is L. Pyber.