Permutation group

Abstract: A finite transitive permutation group is said to be elusive if it has no fixed-point free elements of prime order. In this paper we show that all elusive groups G ¼ N z G1 with N an elementary abelian minimal normal subgroup and G1 cyclic, can be constructed from transitive subgroups of AGLð1; p 2 Þ, for p a Mersenne prime, acting on the set of pð p þ 1Þ lines of the a‰ne plane AGð2; pÞ.

Abstract: Loading takes about thirteen minutes, half that time due to builtin pauses. summary The set of all permutations of a set is a group under composition. binary closure In this section it is shown that bij[x, x] is binary closed under COMPOSE. Lemma.

Abstract: 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: Theorem 2.1 . 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 n ⁄ 2 . Theorem 2.2 . 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 n . 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| = 4 and G is PSL(2,5) permuting the 6 points or the 1-dimensional projective geometry PG(1,5), or |P| = 9 and G is the Mathieu group M 11 in its 3-transitive representation of degree 12. 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. Theorem 2.3 . 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 n . 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. As the first application of these results we prove Theorem 7.1 below about 2-transitive groups of degree r 2 + 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. Theorem 7.1 . If G is a 2-transitive permutation group on Ω of degree n = r 2 + 3r + 3, where r is a prime greater than 3, and such that r divides |G|, then either G contains the alternating group A n , or r is of the form 2 m - 1, a Mersenne prime, for some odd prime m, and G is such that PSL(3,2 m ) ≤ G ≤ PΓL(3,2 m ). Next we turn to 2-transitive groups of degree p 2 , 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 |P| ⁄ 2 . To deal with this, the next theorem, an extension of a result of Felt, was proved. Theorem 9.2 . Let G be a finite group with a cyclic Sylow p-subgroup P of order p k ≥ p 2 , 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 k , such that P is not contained in the kernel of ℒ. Then ℒ p is indecomposable, C G (P) = PxZ(G), and d ≥ (p k +1) ⁄ 2 . Finally there are some results about 2-transitive groups of degree p 2 , following on from Wielendt's classification of the simply transitive groups: Theorem 12.3 . If G is a 2-transitive group of degree p 2 and P is a Sylow p-subgroup of G, then either |P| ≥ p 4 and G contains A p 2 , for p ≥ 3, or |P| = p 3 and G ≤ Aff(2,p), (and G has PSL(2,p) as a composition factor), or |P| = 3 3 and G is PΓL(2,8) of degree 9, or |P| = 2 3 and G is S 4 of degree 4, or |P| = p 2 . If G is primitive of degree p k 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).

Abstract: A file protection scheme for fixed content in a distributed data archive uses computations that leverage permutation operators of a cyclic code. In an illustrative embodiment, an N+K coding technique is described for use to protect data that is being distributed in a redundant array of independent nodes (RAIN). The data itself may be of any type, and it may also include system metadata. According to the invention, the data to be distributed is encoded by a dispersal operation that uses a group of permutation ring operators. In a preferred embodiment, the dispersal operation is carried out using a matrix of the form [I N — C] where I N is an n×n identity sub-matrix and C is a k×n sub-matrix of code blocks. The identity sub-matrix is used to preserve the data blocks intact. The sub-matrix C preferably comprises a set of permutation ring operators that are used to generate the code blocks. The operators are preferably superpositions that are selected from a group ring of a permutation group with base ring Z 2 .