The number of conjugacy classes in pattern groups is not a polynomial function

Abstract

Abstract A famous open problem due to Graham Higman asks if the number of conjugacy classes in the group of n × n unipotent upper triangular matrices over the q-element field can be expressed as a polynomial function of q for every fixed n. We consider the generalization of the problem for pattern groups and prove that for some pattern groups of nilpotency class two the number of conjugacy classes is not a polynomial function of q.