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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$T\hbox {-}groups$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mtext>-</mtext> <mi>g</mi> <mi>r</mi> <mi>o</mi> <mi>u</mi> <mi>p</mi> <mi>s</mi> </mrow> </math></EquationSource> </InlineEquation>). 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 <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$T_c\hbox {-}property$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>T</mi> <mi>c</mi> </msub> <mtext>-</mtext> <mi>p</mi> <mi>r</mi> <mi>o</mi> <mi>p</mi> <mi>e</mi> <mi>r</mi> <mi>t</mi> <mi>y</mi> </mrow> </math></EquationSource> </InlineEquation>), 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 <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$T_c$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>T</mi> <mi>c</mi> </msub> </math></EquationSource> </InlineEquation>-property is metabelian, hypercyclic and abelian-by-finite (see Theorem <InternalRef RefID="FPar1">A</InternalRef>), and that the <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$T_c$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>T</mi> <mi>c</mi> </msub> </math></EquationSource> </InlineEquation>-property is inherited by finite-index subgroups (see Theorem <InternalRef RefID="FPar3">B</InternalRef>). Also, we show that for (affine) algebraic groups, the T-property coincides with the <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$T_c$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>T</mi> <mi>c</mi> </msub> </math></EquationSource> </InlineEquation>-property (see Theorem <InternalRef RefID="FPar5">D</InternalRef>). Note that some of our results are carried out in a more general context by studying a subgroup <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\omega _c(G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ω</mi> <mi>c</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> measuring the distance of a linear group G from having the <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$T_c$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>T</mi> <mi>c</mi> </msub> </math></EquationSource> </InlineEquation>-property. The relationship of <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\omega _c(G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ω</mi> <mi>c</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> with the Wielandt subgroup is studied, and many examples are provided to show how different the behaviour of a linear group with property <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$T_c$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>T</mi> <mi>c</mi> </msub> </math></EquationSource> </InlineEquation> can be from that of a T-group.

Linear groups whose closed subnormal subgroups are normal

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2025年11月15日

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M. Trombetti,
B. A. F. Wehrfritz

摘要

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 $T\hbox {-}groups$ $T - g r o u p s$ ). 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 $T_c\hbox {-}property$ $T_{c} - p r o p e r t y$ ), 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 $T_c$ $T_{c}$ -property is metabelian, hypercyclic and abelian-by-finite (see Theorem A), and that the $T_c$ $T_{c}$ -property is inherited by finite-index subgroups (see Theorem B). Also, we show that for (affine) algebraic groups, the T-property coincides with the $T_c$ $T_{c}$ -property (see Theorem D). Note that some of our results are carried out in a more general context by studying a subgroup $\omega _c(G)$ $ω_{c} (G)$ measuring the distance of a linear group G from having the $T_c$ $T_{c}$ -property. The relationship of $\omega _c(G)$ $ω_{c} (G)$ with the Wielandt subgroup is studied, and many examples are provided to show how different the behaviour of a linear group with property $T_c$ $T_{c}$ can be from that of a T-group.