Linear Groups with Restricted Conjugacy Classes

Abstract

<jats:p>In this paper we characterize, in terms of their conjugacy classes, linear groups <jats:italic>G</jats:italic> such that <jats:inline-formula><jats:alternatives><jats:tex-math>$$G/\\zeta _k(G)$$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mrow>\n <mml:mi>G</mml:mi>\n <mml:mo>/</mml:mo>\n <mml:msub>\n <mml:mi>ζ</mml:mi>\n <mml:mi>k</mml:mi>\n </mml:msub>\n <mml:mrow>\n <mml:mo>(</mml:mo>\n <mml:mi>G</mml:mi>\n <mml:mo>)</mml:mo>\n </mml:mrow>\n </mml:mrow>\n </mml:math></jats:alternatives></jats:inline-formula> belongs to a certain group class <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathfrak {X}$$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mi>X</mml:mi>\n </mml:math></jats:alternatives></jats:inline-formula> for several natural choices of <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathfrak {X}$$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mi>X</mml:mi>\n </mml:math></jats:alternatives></jats:inline-formula>. Moreover, a description is given of linear groups with restrictions on layers.</jats:p>