By C. M. Campbell, M. R. Quick, E. F. Robertson, G. C. Smith

'Groups St Andrews 2005' used to be held within the college of St Andrews in August 2005 and this primary quantity of a two-volume booklet includes chosen papers from the foreign convention. 4 major lecture classes got on the convention, and articles in keeping with their lectures shape a considerable a part of the court cases. This quantity comprises the contributions by way of Peter Cameron (Queen Mary, London) and Rostislav Grogorchuk (Texas A&M, USA). except the most audio system, refereed survey and learn articles have been contributed via different convention members. prepared in alphabetical order, those articles disguise a large spectrum of recent crew conception. The normal lawsuits of teams St Andrews meetings have supplied snapshots of the nation of study in crew conception during the prior 25 years. past volumes have had a big impression at the improvement of crew thought and it truly is expected that this quantity could be both vital.

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**Example text**

4n 3 What happens for primitive groups? The ﬁrst result asserts the existence of a wide gap, between constant and exponential growth. /p(n) for some polynomial p. 148 . .. 324 . .. No counterexamples are known to the conjecture that both parts hold with c = 2. The growth rates of the sequences (fn ) and (Fn ) appear to have a great deal of structure; little is known, but examples suggest some conjectures. One of the most challenging conjectures is to prove that various ‘obvious’ limits, such as lim(log fn )/(log n) (for polynomial growth), lim(log log fn )/(log n) (for fractional exponential growth), or lim(log fn )/n (for exponential growth), actually exist.

Let G1 and G2 be permutation groups on Ω1 and Ω2 respectively. The direct product G1 × G2 has two natural actions: • the intransitive action on the disjoint union of Ω1 and Ω2 , where α(g1 , g2 ) = αgi if α ∈ Ωi , i = 1, 2. • the product action on the Cartesian product of Ω1 and Ω2 , where (α1 , α2 )(g1 , g2 ) = (α1 g1 , α2 g2 ). If G1 and G2 are oligomorphic, then G1 × G2 is oligomorphic in each of these actions. For the intransitive action, things are simple; the modiﬁed cycle index is multiplicative: ˜ 1 × G2 ) = Z(G ˜ 1 )Z(G ˜ 2 ).

Now if G is a permutation group on Ω, then there is a natural action of G on Vn , and hence on A; we let VnG AG = n≥0 be the algebra of A-ﬁxed points. Clearly dim(VnG ) = fn if this number is ﬁnite. The structure of A is known in some cases: 1. If G = S (or indeed if G is highly set-transitive), then AG is the polynomial algebra in a single generator e, the constant function with value 1 in V1 . Cameron: Infinite permutation groups 27 2. If G = G1 × G2 in its intransitive action, then AG = AG1 ⊗C AG2 .