We briefly discuss the algorithm given in [Bates, Bundy, Perkins, Rowley, J. Algebra, 316(2):849868, 2007] for determining the distance between two vertices in a commuting involution graph of a symmetric group.We develop the algorithm in [Bates, Rowley, Arch. Math. (Basel), 85(6):485489, 2005] for computing a subgroup of the normalizer of a 2subgroup X in a finite group G, examining in particular the issue of when to terminate the randomized procedure. The resultant algorithm is capable of handling subgroups X of order up to 512 and is suitable, for example, for matrix groups of large degree (an example calculation is given using 112x112 matrices over GF(2)).We also determine the suborbits of conjugacy classes of involutions in several of the sporadic simple groupsnamely Janko's group J4, the Fischer sporadic groups, and the Thompson and HaradaNorton groups. We use our results to determine the structure of some graphs related to this data.We include implementations of the algorithms discussed in the computer algebra package MAGMA, as well as representative elements for the involution suborbits.
Date of Award  31 Dec 2011 

Original language  English 

Awarding Institution   The University of Manchester


Supervisor  Peter Rowley (Supervisor) & Charles Eaton (Supervisor) 

 Computational group theory
 Finite group theory
Computational Investigation into Finite Groups
Taylor, P. (Author). 31 Dec 2011
Student thesis: Phd