Finite Permutation Groups
<p>The major part of my thesis is concerned with the size and structure of Sylow p-subgroups of a primitive permutation group. The results of Theorems 2.2 and 2.3 were suggested by similar results of Jordan, Manning, Waiss, and othera, about elements of order p in a primitive group. The follow...
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| Médium: | Diplomová práce |
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1973
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| _version_ | 1826263599869329408 |
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| author | Praeger, C |
| author_facet | Praeger, C |
| author_sort | Praeger, C |
| collection | OXFORD |
| description | <p>The major part of my thesis is concerned with the size and structure of Sylow p-subgroups of a primitive permutation group. The results of Theorems 2.2 and 2.3 were suggested by similar results of Jordan, Manning, Waiss, and othera, about elements of order p in a primitive group. The following are the three main results:</p> <p><em>Theorem 2.1</em>. If G is a transitive permutation group on a set Ω of degree n, and if P is a Sylow p-subgroup of G for some prime p dividing |G|, then the number of points of Ω fixed by P is less than <sup>n</sup>⁄<sub>2</sub>.</p> <p><em>Theorem 2.2</em>. Let G be a primitive permutation group on Ω of degree n = kp, where p is a prime, and such that G does not contain the alternating group A<sub>n</sub>. Let P be a Sylow p-subgroup of G, and suppose that P has no orbits of length greater thin p. Then P has order p unless</p> <ol type="a"> <li>|P| = 4 and G is PSL(2,5) permuting the 6 points or the 1-dimensional projective geometry PG(1,5), or</li> <li>|P| = 9 and G is the Mathieu group M<sub>11</sub> in its 3-transitive representation of degree 12.</li> </ol> <p>This result is due to L. Scott for the case in which G is not 2-transitive and my contribution is the 2-transitive case.</p> <p><em>Theorem 2.3</em>. Let G be a 2-transitive permutation group on Ω of degree n = kp + f, for some prime p, such that G does not contain the alternating group A<sub>n</sub>. Suppose that p divides |G| and that a Sylow p-subgroup P of G has k orbits of length p and f fixed points in Ω. Then P has order p unless f = 0.</p> <p>As the first application of these results we prove Theorem 7.1 below about 2-transitive groups of degree r<sup>2</sup> + 3r + 3, where r is a prime. This problem arose from a conjecture about transitive groups of prime degree, and work of Peter Neumann and Tom McDonough.</p> <p><em>Theorem 7.1</em>. If G is a 2-transitive permutation group on Ω of degree n = r<sup>2</sup> + 3r + 3, where r is a prime greater than 3, and such that r divides |G|, then either G contains the alternating group A<sub>n</sub>, or r is of the form 2<sup>m</sup> - 1, a Mersenne prime, for some odd prime m, and G is such that PSL(3,2<sup>m</sup>) ≤ G ≤ PΓL(3,2<sup>m</sup>).</p> <p>Next we turn to 2-transitive groups of degree p<sup>2</sup>, where p is a prime. In looking at the case whore the Sylow p-subgroups are cyclic, the situation arose in which G had an indecomposable representation of degree less than <sup>|P|</sup>⁄<sub>2</sub>. To deal with this, the next theorem, an extension of a result of Felt, was proved.</p> <p><em>Theorem 9.2</em>. Let G be a finite group with a cyclic Sylow p-subgroup P of order p<sup>k</sup> ≥ p<sup>2</sup>, which is a T.I. set. Suppose that G is not p-soluble. Suppose that G has an indecomposable representation ℒ in a field K of characteristic p of degree d ≤ p<sup>k</sup>, such that P is not contained in the kernel of ℒ. Then ℒ<sub>p</sub> is indecomposable, C<sub>G</sub>(P) = PxZ(G), and d ≥ <sup>(p<sup>k</sup>+1)</sup>⁄<sub>2</sub>.</p> <p>Finally there are some results about 2-transitive groups of degree p<sup>2</sup>, following on from Wielendt's classification of the simply transitive groups:</p> <p><em>Theorem 12.3</em>. If G is a 2-transitive group of degree p<sup>2</sup> and P is a Sylow p-subgroup of G, then either</p> <ol type="a"> <li>|P| ≥ p<sup>4</sup> and G contains A<sub>p<sup>2</sup></sub>, for p ≥ 3, or</li> <li>|P| = p<sup>3</sup> and G ≤ Aff(2,p), (and G has PSL(2,p) as a composition factor), or</li> <li>|P| = 3<sup>3</sup> and G is PΓL(2,8) of degree 9, or</li> <li>|P| = 2<sup>3</sup> and G is S<sub>4</sub> of degree 4, or</li> <li>|P| = p<sup>2</sup>.</li> </ol> <p>If G is primitive of degree p<sup>k</sup> and its Sylow p-subgroups are cyclic, we use Theorem 9.2 to extend results of Neumann and Ito, (Theorem 14.2, and Corollary 14.3).</p> |
| first_indexed | 2024-03-06T19:54:21Z |
| format | Thesis |
| id | oxford-uuid:250cc94b-d676-4543-96e9-62d7c8fcc9a2 |
| institution | University of Oxford |
| last_indexed | 2024-03-06T19:54:21Z |
| publishDate | 1973 |
| record_format | dspace |
| spelling | oxford-uuid:250cc94b-d676-4543-96e9-62d7c8fcc9a22022-03-26T11:53:33ZFinite Permutation GroupsThesishttp://purl.org/coar/resource_type/c_db06uuid:250cc94b-d676-4543-96e9-62d7c8fcc9a2Polonsky Theses Digitisation Project1973Praeger, C<p>The major part of my thesis is concerned with the size and structure of Sylow p-subgroups of a primitive permutation group. The results of Theorems 2.2 and 2.3 were suggested by similar results of Jordan, Manning, Waiss, and othera, about elements of order p in a primitive group. The following are the three main results:</p> <p><em>Theorem 2.1</em>. If G is a transitive permutation group on a set Ω of degree n, and if P is a Sylow p-subgroup of G for some prime p dividing |G|, then the number of points of Ω fixed by P is less than <sup>n</sup>⁄<sub>2</sub>.</p> <p><em>Theorem 2.2</em>. Let G be a primitive permutation group on Ω of degree n = kp, where p is a prime, and such that G does not contain the alternating group A<sub>n</sub>. Let P be a Sylow p-subgroup of G, and suppose that P has no orbits of length greater thin p. Then P has order p unless</p> <ol type="a"> <li>|P| = 4 and G is PSL(2,5) permuting the 6 points or the 1-dimensional projective geometry PG(1,5), or</li> <li>|P| = 9 and G is the Mathieu group M<sub>11</sub> in its 3-transitive representation of degree 12.</li> </ol> <p>This result is due to L. Scott for the case in which G is not 2-transitive and my contribution is the 2-transitive case.</p> <p><em>Theorem 2.3</em>. Let G be a 2-transitive permutation group on Ω of degree n = kp + f, for some prime p, such that G does not contain the alternating group A<sub>n</sub>. Suppose that p divides |G| and that a Sylow p-subgroup P of G has k orbits of length p and f fixed points in Ω. Then P has order p unless f = 0.</p> <p>As the first application of these results we prove Theorem 7.1 below about 2-transitive groups of degree r<sup>2</sup> + 3r + 3, where r is a prime. This problem arose from a conjecture about transitive groups of prime degree, and work of Peter Neumann and Tom McDonough.</p> <p><em>Theorem 7.1</em>. If G is a 2-transitive permutation group on Ω of degree n = r<sup>2</sup> + 3r + 3, where r is a prime greater than 3, and such that r divides |G|, then either G contains the alternating group A<sub>n</sub>, or r is of the form 2<sup>m</sup> - 1, a Mersenne prime, for some odd prime m, and G is such that PSL(3,2<sup>m</sup>) ≤ G ≤ PΓL(3,2<sup>m</sup>).</p> <p>Next we turn to 2-transitive groups of degree p<sup>2</sup>, where p is a prime. In looking at the case whore the Sylow p-subgroups are cyclic, the situation arose in which G had an indecomposable representation of degree less than <sup>|P|</sup>⁄<sub>2</sub>. To deal with this, the next theorem, an extension of a result of Felt, was proved.</p> <p><em>Theorem 9.2</em>. Let G be a finite group with a cyclic Sylow p-subgroup P of order p<sup>k</sup> ≥ p<sup>2</sup>, which is a T.I. set. Suppose that G is not p-soluble. Suppose that G has an indecomposable representation ℒ in a field K of characteristic p of degree d ≤ p<sup>k</sup>, such that P is not contained in the kernel of ℒ. Then ℒ<sub>p</sub> is indecomposable, C<sub>G</sub>(P) = PxZ(G), and d ≥ <sup>(p<sup>k</sup>+1)</sup>⁄<sub>2</sub>.</p> <p>Finally there are some results about 2-transitive groups of degree p<sup>2</sup>, following on from Wielendt's classification of the simply transitive groups:</p> <p><em>Theorem 12.3</em>. If G is a 2-transitive group of degree p<sup>2</sup> and P is a Sylow p-subgroup of G, then either</p> <ol type="a"> <li>|P| ≥ p<sup>4</sup> and G contains A<sub>p<sup>2</sup></sub>, for p ≥ 3, or</li> <li>|P| = p<sup>3</sup> and G ≤ Aff(2,p), (and G has PSL(2,p) as a composition factor), or</li> <li>|P| = 3<sup>3</sup> and G is PΓL(2,8) of degree 9, or</li> <li>|P| = 2<sup>3</sup> and G is S<sub>4</sub> of degree 4, or</li> <li>|P| = p<sup>2</sup>.</li> </ol> <p>If G is primitive of degree p<sup>k</sup> and its Sylow p-subgroups are cyclic, we use Theorem 9.2 to extend results of Neumann and Ito, (Theorem 14.2, and Corollary 14.3).</p> |
| spellingShingle | Praeger, C Finite Permutation Groups |
| title | Finite Permutation Groups |
| title_full | Finite Permutation Groups |
| title_fullStr | Finite Permutation Groups |
| title_full_unstemmed | Finite Permutation Groups |
| title_short | Finite Permutation Groups |
| title_sort | finite permutation groups |
| work_keys_str_mv | AT praegerc finitepermutationgroups |