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530 ARCH. MATH. Some permutation groups of degree p=6q+l
 

Summary: 530 ARCH. MATH.
4
Some permutation groups of degree p=6q+l
By
~I. D. ATxi~so~
Abstract. Transitive permutation groups of degrees 43, 67, 79, 103 and 139 are
classified.
In this note we consider insoluble transitive permutation groups of degree p ----
6 q + 1 where p and q are primes and summarise the computations whereby these
groups have been classified for some small values of q. The result which allows
progress on this problem is due to McDonough [1] ; he showed that ff such a group
has a Sylow p-normaliser of order 3p then it is isomorphic either to PSL(3, 3) or
PS.L (3, 5) (of degrees 13, 31 respectively). Using this theorem machine computations
along the lines of those done by Parker, Nikolai and Appel [3, 2] for degrees p ----
2 q + 1 and T -~ 4 q + 1 give the following
Theorem. Every insoluble transitive permutation grouT o/ degree 43, 67, 79, 103, 139
contains the alternating group o/ that degree.
To describe the calculations leading to this result we let G denote an insoluble
transitive group of degree p ~ 6q Jr 1, T and q prime, with q > 5 and let P be
a Sylow p-subgroup of G. In trying to prove that G ~ A~ or S~ we can of course

  

Source: Atkinson, Mike - Department of Computer Science, University of Otago

 

Collections: Computer Technologies and Information Sciences