By Cheryl E. Praeger

ISBN-10: 0511526032

ISBN-13: 9780511526039

ISBN-10: 0521567378

ISBN-13: 9780521567374

This publication addresses yes methods of representing sporadic easy teams and the graphs linked to those representations. It offers descriptions of all permutation representations of rank as much as 5 of the sporadic teams and offers an advent to functional computational recommendations required for learning finite vertex-transitive graphs. The authors supply adequate information regarding lots of the teams to permit readers to reconstruct and examine those representations themselves. undemanding suggestions for research of finite vertex-transitive graphs are rigorously defined, making this ebook not just a vital reference for specialists, but in addition a great place to begin for graduate scholars and researchers during this quarter.

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**Extra resources for Low rank representations and graphs for sporadic groups**

**Sample text**

Then E provides a simple tool for determining whether r is simple, whether r is connected, and if so for finding its diameter, and whether G acts on r distance-transitively. If r is simple and connected, then E together with A provide the tool-kit for determining the girth of r, and whether r is distance-regular. 5 A lemma for certain imprimitive groups The result below will be used to eliminate from consideration certain imprimitive representations which have rank greater than 5. 4 Let G be a transitive imprimitive permutation group on a finite set n, and let q> :::: {B9 I g E G} be a system of blocks of imprimitivity preserved by G.

16], we see that either (a) holds, or r = 5 and 7r = 1a + 2X for some nontrivial irreducible character X. Suppose the latter holds. The character X has algebraic integer values and, since 7r has rational integer values, X too must have rational integer values. Thus, each value of 7r is an odd rational integer, and so positive. 20-21]) [la,7r] is greater than 1, which contradicts the fact that 1a has multiplicity 1 in 7r. 1 a pseudo-permutation character for G (of rank at most 5). We first used GAP to compute a list of all pseudo-permutation characters of rank at most 5 for the sporadic almost simple groups.

Thus we have described, up to permutational isomorphism, a unique imprimitive rank 5 representation with this permutation character. ) 0 5 0 4 5 1) 55 20 20 22 20 30 ) 24 25 3~ 4 44 The Individual Groups o0 o o 0 0 6 6 ( 1 5 0 6 0 6 5 0 30 30 24 25 66 30 30 30 30 1 . The non-complete distance-regular generalized orbital graphs are r{2,3}, r{4}, r{2,4}, r{3,4}, r{5}, r{2,4,5} e:! r{3, 4, 5} e:! = t {22, 11j 1, 2} t = {55, 32j 1, 20} t= {66,35j1,30} t= {66,35j1,30} t= {66,35j1,30} (and (and (and (and (and complement) complement) complement) complement) complement) Kl2X12 K12X12.

### Low rank representations and graphs for sporadic groups by Cheryl E. Praeger

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