Noether’s Problem for p-Groups with Abelian Normal Subgroups and Central p-Powers

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

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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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author	Ivo M. Michailov Ivailo A. Dimitrov
author_facet	Ivo M. Michailov Ivailo A. Dimitrov
author_sort	Ivo M. Michailov
collection	DOAJ
description	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.
format	Article
id	doaj-art-33a823e8d4b94cebacdb64f45ab4fc9f
institution	Kabale University
issn	2227-7390
language	English
publishDate	2024-12-01
publisher	MDPI AG
record_format	Article
series	Mathematics
spelling	doaj-art-33a823e8d4b94cebacdb64f45ab4fc9f2025-01-10T13:18:07ZengMDPI AGMathematics2227-73902024-12-011315610.3390/math13010056Noether’s Problem for <i>p</i>-Groups with Abelian Normal Subgroups and Central <i>p</i>-PowersIvo M. Michailov0Ivailo A. Dimitrov1Faculty of Mathematics and Informatics, Paisii Hilendarski University of Plovdiv, 24 Tsar Assen Str., 4027 Plovdiv, BulgariaPMG Nancho Popovich, Str. Kiril I Metodiy 68, 9700 Shumen, BulgariaThis 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.https://www.mdpi.com/2227-7390/13/1/56Noether’s problemthe rationality problemnilpotent groupsp-groupsmetabelian groups
spellingShingle	Ivo M. Michailov Ivailo A. Dimitrov Noether’s Problem for <i>p</i>-Groups with Abelian Normal Subgroups and Central <i>p</i>-Powers Mathematics Noether’s problem the rationality problem nilpotent groups p-groups metabelian groups
title	Noether’s Problem for <i>p</i>-Groups with Abelian Normal Subgroups and Central <i>p</i>-Powers
title_full	Noether’s Problem for <i>p</i>-Groups with Abelian Normal Subgroups and Central <i>p</i>-Powers
title_fullStr	Noether’s Problem for <i>p</i>-Groups with Abelian Normal Subgroups and Central <i>p</i>-Powers
title_full_unstemmed	Noether’s Problem for <i>p</i>-Groups with Abelian Normal Subgroups and Central <i>p</i>-Powers
title_short	Noether’s Problem for <i>p</i>-Groups with Abelian Normal Subgroups and Central <i>p</i>-Powers
title_sort	noether s problem for i p i groups with abelian normal subgroups and central i p i powers
topic	Noether’s problem the rationality problem nilpotent groups p-groups metabelian groups
url	https://www.mdpi.com/2227-7390/13/1/56
work_keys_str_mv	AT ivommichailov noethersproblemforipigroupswithabeliannormalsubgroupsandcentralipipowers AT ivailoadimitrov noethersproblemforipigroupswithabeliannormalsubgroupsandcentralipipowers

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

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