TY - JOUR

T1 - Finite skew braces with solvable additive group

AU - Gorshkov, Ilya

AU - Nasybullov, Timur

N1 - Funding Information: Acknowledgment. The work is supported by Mathematical Center in Akademgorodok under agreement No. 075-15-2019-1613 with the Ministry of Science and Higher Education of the Russian Federation. Publisher Copyright: © 2021 Elsevier Inc. Copyright: Copyright 2021 Elsevier B.V., All rights reserved.

PY - 2021/5/15

Y1 - 2021/5/15

N2 - A. Smoktunowicz and L. Vendramin conjectured that if A is a finite skew brace with solvable additive group, then the multiplicative group of A is solvable. In this short note we make a step towards positive solution of this conjecture proving that if A is a minimal finite skew brace with solvable additive group and non-solvable multiplicative group, then the multiplicative group of A is not simple. On the way to obtaining this result, we prove that the conjecture of A. Smoktunowicz and L. Vendramin is correct in the case when the order of A is not divisible by 3.

AB - A. Smoktunowicz and L. Vendramin conjectured that if A is a finite skew brace with solvable additive group, then the multiplicative group of A is solvable. In this short note we make a step towards positive solution of this conjecture proving that if A is a minimal finite skew brace with solvable additive group and non-solvable multiplicative group, then the multiplicative group of A is not simple. On the way to obtaining this result, we prove that the conjecture of A. Smoktunowicz and L. Vendramin is correct in the case when the order of A is not divisible by 3.

KW - Simple group

KW - Skew brace

KW - Solvable group

UR - http://www.scopus.com/inward/record.url?scp=85100437429&partnerID=8YFLogxK

U2 - 10.1016/j.jalgebra.2021.01.027

DO - 10.1016/j.jalgebra.2021.01.027

M3 - Article

AN - SCOPUS:85100437429

VL - 574

SP - 172

EP - 183

JO - Journal of Algebra

JF - Journal of Algebra

SN - 0021-8693

ER -