Geodesic
On automorphism groups of endomorphism semigroups of finite elementary Abelian groups
Proceedings of the Yerevan State University. Physical and mathematical sciences, Tome 56 (2022) no. 2, pp. 49-57.

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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. 
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A. A. Bayramyan. On automorphism groups of endomorphism semigroups of finite elementary Abelian groups. Proceedings of the Yerevan State University. Physical and mathematical sciences, Tome 56 (2022) no. 2, pp. 49-57. http://geodesic.mathdoc.fr/item/UZERU_2022_56_2_a1/

[1] B. I. Plotkin, Seven Lectures on the Universal Algebraic Geometry, Preprint, 2002, 87 pp., arXiv: math/0204245[math.GM] | MR

[2] E. Formanek, “A Question of B. Plotkin about the Semigroup of Endomorphisms of a Free Group”, Proc. American Math. Society, 130 (2001), 935–937 | DOI | MR | Zbl

[3] V. S. Atabekyan, “The Automorphisms of Endomorphism Semigroups of Free Burnside Groups”, Intern. J. Algebra and Computation, 25:04 (2015), 669–674 | DOI | MR | Zbl

[4] V. S. Atabekyan, H. T. Aslanyan, “The Automorphisms of Endomorphism Semigroups of Relatively Free Groups”, International Journal of Algebra and Computation, 28 (2018), 207–215 | DOI | MR | Zbl

[5] L. M. Gluskin, “Automorphisms of Multiplicative Semigroups of Matrix Algebras”, Uspehi Mat. Nauk (N.S.), 11:1 (1956), 199–206 (in Russian) | MR | Zbl

[6] E. A. Halezov, “Automorphisms of Matrix Semigroups”, Dokl. Akad. Nauk SSSR, 96:2 (1954), 245–248 (in Russian) | MR | Zbl

[7] W. C. Waterhouse, “Two Generators for the General Linear Groups over Finite Fields”, Linear Multilinear Algebra, 24 (1988), 227–230 | DOI | MR

[8] J. Dieudonne, “On the Automorphisms of the Classical Groups”, Mem. Amer. Math. Soc., 2 (1951), 1–95 | DOI | MR

[9] W. C. Waterhouse, “Automorphisms of $\mathrm{GL}_n(R)$”, Proc. Am. Math. Soc., 79:3 (1980), 347–351 | DOI | MR | Zbl