A Generalization of Sylow’s Theorem

A Generalization of Sylow’s Theorem

Bibliographic Details
Main Author: Thomas, Teri M.
Language:English
Published: Youngstown State University / OhioLINK 2009
Subjects:
Mathematics
Hall&8217
s theorem
Sylow&8217
group theory
abstract algebra
Online Access:http://rave.ohiolink.edu/etdc/view?acc_num=ysu1256911896
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spelling ndltd-OhioLink-oai-etd.ohiolink.edu-ysu12569118962021-08-03T06:17:54Z A Generalization of Sylow’s Theorem Thomas, Teri M. Mathematics Hall&8217 s theorem Sylow&8217 s theorem group theory abstract algebra In the study of group theory, it is common to break up a complex group into simpler subgroups in order to arrive at a structure that is easier to analyze and understand. It is also sometimes possible to reconstruct the original group from these subgroups. Although this is not always possible,we can apply this process to finite solvable groups and derive some theorems regarding these groups. Sylow’s Theorem and Hall’s Theorem are among the most famous results. Hall’s Theorem, which is regarded as an extension of Sylow’s Theorem, states that if a group <i>G</i> is solvable and is of some order <i>mn</i>, where <i>m</i> is prime to <i>n</i>, then <i>G</i> has a subgroup of order <i>m</i> and all subgroups of this order are conjugate. When <i>p</i> = π, a Hall π-subgroup is simply a Sylow <i>p</i>-subgroup. While Sylow’s Theorem is valid for any finite group, Hall subgroups need not exist in nonsolvablegroups. For example, A<sub>5</sub> has order 60 = 3 ⋅ 20, but it has no subgroups of order 20. This is demonstrated within the paper. Hall’s Theorem has been the starting point for the theory of finite solvable groups developed over the past seventy years, although those results are not given here. 2009-10-30 English text Youngstown State University / OhioLINK http://rave.ohiolink.edu/etdc/view?acc_num=ysu1256911896 http://rave.ohiolink.edu/etdc/view?acc_num=ysu1256911896 unrestricted This thesis or dissertation is protected by copyright: all rights reserved. It may not be copied or redistributed beyond the terms of applicable copyright laws.
collection NDLTD
language English
sources NDLTD
topic Mathematics
Hall&8217
s theorem
Sylow&8217
s theorem
group theory
abstract algebra
spellingShingle Mathematics
Hall&8217
s theorem
Sylow&8217
s theorem
group theory
abstract algebra
Thomas, Teri M.
A Generalization of Sylow’s Theorem
author Thomas, Teri M.
author_facet Thomas, Teri M.
author_sort Thomas, Teri M.
title A Generalization of Sylow’s Theorem
title_short A Generalization of Sylow’s Theorem
title_full A Generalization of Sylow’s Theorem
title_fullStr A Generalization of Sylow’s Theorem
title_full_unstemmed A Generalization of Sylow’s Theorem
title_sort generalization of sylow’s theorem
publisher Youngstown State University / OhioLINK
publishDate 2009
url http://rave.ohiolink.edu/etdc/view?acc_num=ysu1256911896
work_keys_str_mv AT thomasterim ageneralizationofsylowstheorem
AT thomasterim generalizationofsylowstheorem
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