Dear all,
I've contributed a too trivial question today, hopefully this time it is not that trivial...
Let A be a finite abelian p-group, and R is the endomorphism ring of A. Define the operation of R "*" as follows: M*N=M+N+pMN. It can be verified that (R, *) is also a group (not necessarily commutative).
Now assume that S is a subgroup of (R, +) and (R, *) at the same time, and assume that M is a set of generators of (S, +). Can we recover a set of generators of (S, *)?
A natural example of S is the centralizer of an element in R, that is for N in R, S={M in R, MN=NM}. Solving a system of Diophantine equations gives a set of generators of (S, +), but we need a set of generators for (S, *).
Best,
Youming
