Abstract:
In this article, we explore the automorphisms of endomorphism semigroups and automorphism groups of the finite elementary Abelian groups. In particular, we prove that $\mathrm{Aut}(\mathrm{End}(\mathbb{Z}_p\oplus\mathbb{Z}_p\oplus\cdots\oplus\mathbb{Z}_p))$ can be canonically embedded into $\mathrm{Aut}(\mathrm{Aut}(\mathbb{Z}_p\oplus\mathbb{Z}_p\oplus\cdots\oplus\mathbb{Z}_p))$ using an elementary approach based on matrix operations. We also show that all automorphisms of $\mathrm{End}(\mathbb{Z}_p\oplus\mathbb{Z}_p\oplus\cdots\oplus\mathbb{Z}_p)$ are inner.
Keywords:automorphisms of matrix semigroups, finite elementary Abelian groups, automorphisms of endomorphism semigroup.