On the groups associated with a tropical n × n matrix

In this paper, the generalized centralizer group Un(A) of a tropical n×n matrix A and the centralizer group Pn(E) of a tropical idempotent normal matrix E are introduced and studied. It is proved that Un(A) is a product of two specific normal subgroups. And a structural description of Pn(E) is given when E is not strongly regular. It is also made some observations on E when Pn(E) is isomorphic to a 2-closed transitive permutation group on {1,2,…,n}.