Noether’s Problem for <i>p</i>-Groups with Abelian Normal Subgroups and Central <i>p</i>-Powers

Noether’s Problem for <i>p</i>-Groups with Abelian Normal Subgroups and Central <i>p</i>-Powers

This paper addresses Noether’s problem for <i>p</i>-groups <i>G</i>, having an abelian normal subgroup of index <i>p</i>, under the condition <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><sema...

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Bibliographic Details
Main Authors:	Ivo M. Michailov, Ivailo A. Dimitrov
Format:	Article
Language:	English
Published:	MDPI AG 2024-12-01
Series:	Mathematics
Subjects:	Noether’s problem the rationality problem nilpotent groups p-groups metabelian groups
Online Access:	https://www.mdpi.com/2227-7390/13/1/56
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Summary:	This paper addresses Noether’s problem for <i>p</i>-groups <i>G</i>, having an abelian normal subgroup of index <i>p</i>, under the condition <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>G</mi><mi>p</mi></msup><mo>=</mo><mrow><mo>{</mo><msup><mi>g</mi><mi>p</mi></msup><mo>:</mo><mi>g</mi><mo>∈</mo><mi>G</mi><mo>}</mo></mrow><mo>≤</mo><mi>Z</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>—the center of <i>G</i>. We prove that the fixed field <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>K</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mi>K</mi><msup><mrow><mo>(</mo><mi>x</mi><mrow><mo>(</mo><mi>g</mi><mo>)</mo></mrow><mo>:</mo><mi>g</mi><mo>∈</mo><mi>G</mi><mo>)</mo></mrow><mi>G</mi></msup></mrow></semantics></math></inline-formula> is rational over <i>K</i> in such cases, focusing on both the classification and structural analysis of these groups. Our results extend existing work by removing restrictive assumptions and providing a refined understanding of <i>p</i>-groups and their representations.
ISSN:	2227-7390