Linear groups whose closed subnormal subgroups are normal

Abstract Although normality is not a transitive relation in group theory, one often needs to deal with sections of a group in which this is the case. This led many prominent group theorists, such as Gaschütz and Robinson, to study (finite and infinite) groups in which normality is transitive (the so-called $$T\hbox {-}groups$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>T</mml:mi> <mml:mtext>-</mml:mtext> <mml:mi>g</mml:mi> <mml:mi>r</mml:mi> <mml:mi>o</mml:mi> <mml:mi>u</mml:mi> <mml:mi>p</mml:mi> <mml:mi>s</mml:mi> </mml:mrow> </mml:math> ). The aim of this paper is to describe (mostly periodic soluble) linear groups in which normality of Zariski closed subgroups is transitive (we call this property the $$T_c\hbox {-}property$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>c</mml:mi> </mml:msub> <mml:mtext>-</mml:mtext> <mml:mi>p</mml:mi> <mml:mi>r</mml:mi> <mml:mi>o</mml:mi> <mml:mi>p</mml:mi> <mml:mi>e</mml:mi> <mml:mi>r</mml:mi> <mml:mi>t</mml:mi> <mml:mi>y</mml:mi> </mml:mrow> </mml:math> ), so to obtain a more general framework from which some of the well-known and relevant results concerning finite soluble T -groups can be derived. Our main theorems show among other things that a soluble periodic linear group with $$T_c$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>c</mml:mi> </mml:msub> </mml:math> -property is metabelian, hypercyclic and abelian-by-finite (see Theorem A), and that the $$T_c$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>c</mml:mi> </mml:msub> </mml:math> -property is inherited by finite-index subgroups (see Theorem B). Also, we show that for (affine) algebraic groups, the T -property coincides with the $$T_c$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>c</mml:mi> </mml:msub> </mml:math> -property (see Theorem D). Note that some of our results are carried out in a more general context by studying a subgroup $$\omega _c(G)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>ω</mml:mi> <mml:mi>c</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>G</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> measuring the distance of a linear group G from having the $$T_c$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>c</mml:mi> </mml:msub> </mml:math> -property. The relationship of $$\omega _c(G)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>ω</mml:mi> <mml:mi>c</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>G</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> with the Wielandt subgroup is studied, and many examples are provided to show how different the behaviour of a linear group with property $$T_c$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>c</mml:mi> </mml:msub> </mml:math> can be from that of a T -group.