Sylow subgroups of index 2 in their normalizers
<p>The following theorem is proved:</p> <p><strong>Theorem</strong> Let G be a finite group, P a Sylow p-subgroup of G, p odd. Suppose</p> <ol type="1"> <li>|N(P)/P⋅C(P)| = 2;</li> <li>cl(P) ≤ 2 .</li> </ol> <p>T...
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| Format: | Thesis |
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1973
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| _version_ | 1797063136970276864 |
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| author | Smith, S |
| author_facet | Smith, S |
| author_sort | Smith, S |
| collection | OXFORD |
| description | <p>The following theorem is proved:</p> <p><strong>Theorem</strong> Let G be a finite group, P a Sylow p-subgroup of G, p odd. Suppose</p> <ol type="1"> <li>|N(P)/P⋅C(P)| = 2;</li> <li>cl(P) ≤ 2 .</li> </ol> <p>Then</p> <ol type="i"> <li>if G is perfect, then P is necessarily cyclic;</li> <li>if P is not cyclic, then either 0<sup>p</sup>(G) < G, or 0<sup>2</sup>(G) < G with G = 0<sub>p</sub>,(G)⋅N(P).</li> </ol> <p>A unified proof is given as far as possible, but the proof eventually splits into three cases, with hypothesis (2) strengthened by one of:</p> <ol type="A"> <li>|P′| = p,</li> <li>P is abelian but not cyclic, or</li> <li>|P′| > p.</li> </ol> <p>Different methods are in fact required for each case.</p> <p>Several corollaries are also discussed.</p> |
| first_indexed | 2024-03-06T20:55:31Z |
| format | Thesis |
| id | oxford-uuid:391b761e-9802-4fef-819b-512c3daa496e |
| institution | University of Oxford |
| last_indexed | 2024-03-06T20:55:31Z |
| publishDate | 1973 |
| record_format | dspace |
| spelling | oxford-uuid:391b761e-9802-4fef-819b-512c3daa496e2022-03-26T13:53:42ZSylow subgroups of index 2 in their normalizersThesishttp://purl.org/coar/resource_type/c_db06uuid:391b761e-9802-4fef-819b-512c3daa496ePolonsky Theses Digitisation Project1973Smith, S<p>The following theorem is proved:</p> <p><strong>Theorem</strong> Let G be a finite group, P a Sylow p-subgroup of G, p odd. Suppose</p> <ol type="1"> <li>|N(P)/P⋅C(P)| = 2;</li> <li>cl(P) ≤ 2 .</li> </ol> <p>Then</p> <ol type="i"> <li>if G is perfect, then P is necessarily cyclic;</li> <li>if P is not cyclic, then either 0<sup>p</sup>(G) < G, or 0<sup>2</sup>(G) < G with G = 0<sub>p</sub>,(G)⋅N(P).</li> </ol> <p>A unified proof is given as far as possible, but the proof eventually splits into three cases, with hypothesis (2) strengthened by one of:</p> <ol type="A"> <li>|P′| = p,</li> <li>P is abelian but not cyclic, or</li> <li>|P′| > p.</li> </ol> <p>Different methods are in fact required for each case.</p> <p>Several corollaries are also discussed.</p> |
| spellingShingle | Smith, S Sylow subgroups of index 2 in their normalizers |
| title | Sylow subgroups of index 2 in their normalizers |
| title_full | Sylow subgroups of index 2 in their normalizers |
| title_fullStr | Sylow subgroups of index 2 in their normalizers |
| title_full_unstemmed | Sylow subgroups of index 2 in their normalizers |
| title_short | Sylow subgroups of index 2 in their normalizers |
| title_sort | sylow subgroups of index 2 in their normalizers |
| work_keys_str_mv | AT smiths sylowsubgroupsofindex2intheirnormalizers |