Counting nilpotent endomorphisms

Nilpotent Families of Endomorphisms of ...

Rational Endomorphisms of a Nilpotent Group

Let G be a group. An endomorphism φ of G is called rational if there exist a1, . . . , ar ∈ G and h1, . . . , hr ∈ Z, such that φ(x) = (xa1)1 . . . (xar)r for all x ∈ G. We denote by Endr(G) the group of invertible rational endomorphisms of G. In this note, we prove that G is nilpotent of class c (c ≥ 3) if and only if Endr(G) is nilpotent of class c − 1. Mathematics Subject Classification: 20E...

Counting Symmetric Nilpotent Matrices

We determine the number of nilpotent matrices of order n over Fq that are selfadjoint for a given nondegenerate symmetric bilinear form, and in particular find the number of symmetric nilpotent matrices.

Counting Endomorphisms of Crown-like Orders

The authors introduce the notion of crown-like orders and introduce powerful tools for counting the endomorphisms of orders of this type. r r r r r r r r @ @ @ @ @ @ 1 3 5 7 2 4 6 8 r r r r r r r r @ @ @ @ @ @ 3 3 3 3 3 3 3 1 3 5 7 2 4 6 8 Figure 1: The fence and crown of order 4 r r r r r r r r r r r r · · · · · · @ @ @ @ @ @ @ @ @ @ −5 −3 −1 1 3 5 −4 −2 0 2 4 6 Figure 2: The infinite fence F

On the structure of nilpotent endomorphisms and applications

The nilpotent endomorphisms over a finite free module over a domain with principal ideal are characterized. One may apply these results to the study of the maximal Cohen-Macaulay modules over the ring R := A[[x]]/(x), n ≥ 2, where A is a DVR. Subject Classification: 15A21, 13C14.