On the number of conjugacy classes of finite nilpotent groups
We establish the first super-logarithmic lower bound for the number of conjugacy classes of a finite nilpotent group. In particular, we obtain that for any constant c there are only finitely many finite p-groups of order pm with at most c⋅m conjugacy classes. This answers a question of L. Pyber.
Abstract This paper deals with the complexity of context-free grammars with 1-letter terminal alphabet. We study the complexity of the membership problem and the inequivalence problem. We show that the first problem is NP-complete and the second one is Σ P 2 -complete with respect to log-space reduction. The second result also implies that the inequivalence problem is in P space , solving an open problem stated by Hunt, Rosenkrantz and Szymanski (1976).