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Generators of subgroups d G is the maximum of d H over all subgroups H G . n 2 McIver and Neumann showed that d S n for 3 . This

  1. ✁ ✎ ✁ ☎ ✖ ✒ ✒ ✒ ✟ ☎ ✁ ☎ � ✡ ✏ Generators of subgroups d ✟✠✄ G ☎ is the maximum of d ✄ H ☎ over all subgroups H G . ✁☞☛ n ✌ 2 McIver and Neumann showed that d ✄ S n ✍ for 3 . This is a lower bound, as is shown by n How big is the ✄ 1 ✂ 2 ☎✑✂✞✄ 3 ✂ 4 ☎✑✂✞✆✝✆✝✆✝✂✞✄ 2 m 1 ✂ 2 m symmetric group? ☎✝✓ ✁☞☛ n ✌ 2 where m ✍ . Showing it is an upper bound is harder! Peter J Cameron Jerrum gave a more elementary proof that School of Mathematical Sciences 1 . In fact he showed that any subgroup of d ✟✠✄ S n n ☎✔✡ Queen Mary, University of London S n has a “nice” generating set with the properties London E1 4NS, U.K. ✕ a “nice” generating set contains at most n 1 p.j.cameron@qmul.ac.uk elements; PGGT, Birmingham, 17 April 2002 ✕ if S is “nice” and g S n , then we can compute a ✏ S “nice” generating set for ✓ efficiently. ✂ g This can be used to compute a base and strong generating set of an arbitrary subgroup in polynomial time. 1 3 Length of subgroup chain Order, composition factors, generators ☎ is the length of the longest chain of subgroups in l ✄ G The order of S n is n ! (trivial). G . The number of composition factors of S n is 2 for Note that d ☎ for any group G ; this was the ✟✗✄ G l ✄ G 2 ✂ 4 (known to Galois?). ☎✘✡ n original motivation for studying l ☎ (Babai). ✄ G ☎ is the minimum number of generators of the d ✄ G If N is a normal subgroup of G , then 2 (elementary: for group G . We have d ✄ S n ☎ . Thus we only have to compute l ✄ G l ✄ N l ✄ G ✌ N ✄ 1 ✂ 2 ✄ 1 ✂ 2 ☎✚✙ example, ✂✝✆✞✆✝✆✝✂ n ☎ and ☎ generate S n ). ☎ for all simple groups S . l ✄ S So the symmetric group is both very big and very This has been done for many families of simple small! groups (Solomon, Turull). 2 4

  2. ✎ ✒ ✤ ✖ ✌ ☎ ✁ ☎ ✟ ✁ ✒ ✧ ✂ ✖ ✒ ✖ ✧ ✧ ✧ ✁ ✖ ☎ ✖ ✖ ✖ ☎ Independent generating sets Length of subgroup chain 1 . Again, the Whiston proved that µ ✄ S n µ ✭✮✄ S n n Cameron, Solomon and Turull showed (using CFSG) lower bound is straightforward (the set that ✁☞✛ 3 n ✌ 2 1 ✩✯✄ 1 ✂ 2 ☎✰✂✞✄ 2 ✂ 3 1 l ✄ S n b ✄ n ☎✑✂✝✆✞✆✝✆✝✂✞✄ n ✂ n ✜✘✒ ☎✢✒ ☎✱✫ ☎ is the number of ones in the base 2 where b ✄ n is an independent generating set). For the upper 4 . representation of n . So d ✄ S n l ✄ S n ☎ for n bound, CFSG is required; if G is the subgroup ☎✘✣ generated by all but one element of an independent 1 , It is easy to find a subgroup chain whose length is generating set of largest size, then µ ✄ S n µ ✄ G ☎✔✡ ☎✚✙ the right-hand side of the above formula. To show and we have to analyse G . that no longer chain is possible, we take a chain Cameron and Cara used Whiston’s result to S n G ✎✦✥✝✥✝✥ determine all independent generating sets of S n of 1 . There are two types; one consists of and analyse the possibilities for G , ultimately using size n the O’Nan–Scott Theorem and CFSG. transpositions corresponding to the edges of a tree; the other contains one transposition, the other elements being 3 -cycles or double transpositions. Probably the use of CFSG here can be avoided! 5 7 Independent generating sets Coset geometries ✏ S A set S G is independent if s ★✪✩ s ✫✬✓ for all s S ; ✄ G i : i Let G be a group, and I ☎ a family of that is, no element of s can be written as a word in subgroups of G . for J I , let G J ✴ J G j . Suppose the remaining elements. ✁✳✲ j that the following three conditions hold: µ ✭✬✄ G ☎ is the size of the largest independent subset of G ; and µ ☎ is the size of the largest independent ✄ G (G1) The subgroups G J , for J I , are all distinct. generating set of G . Clearly µ ✄ G µ ✭✬✄ G ☎ . Equality ☎✘✡ holds in abelian groups, p -groups, dihedral groups, 1 , then (G2) If J I and ✵ J ✵✶✣✷✵ I ✏ G J and (as we will see) symmetric groups, but not in ✵✝✒ ✻ : k G J I ★ J ✓ . general. (Whiston gives counterexamples in PSL ✄ 2 ✸✺✹ k ✂ p for suitable primes p .) It would be interesting to know ✄ G j x j : j (G3) If a family ☎ of right cosets have J more about this! Also, µ ✄ G ☎ is the size of the largest pairwise non-empty intersection, for j J , then there minimal (w.r.t. inclusion) generating set of G . is an element of G lying in all these cosets. The parameter µ ☎ occurs in the paper of Diaconis ✄ G and Saloff-Coste on the rate of convergence of the ✂✱✄ G i : i The coset geometry C ☎✝☎ has type set I ; ✄ G I product replacement algorithm on a finite group. The the varieties of type i are the right cosets of G i , and time required until the distribution is near-random two varieties are incident if their intersection is depends very sensitively on µ ☎ . ✄ G non-empty. The group G acts as a flag-transitive automorphism group of the geometry. Another application is given later. 6 8

  3. ☎ ✒ ✒ ✼ ✖ ☎ ✁ ✭ Coset geometries Other measures? The coset geometry is called residually weakly primitive, or RWPri, if the following condition holds: Both l ✄ G ☎ and µ ✭✬✄ G ☎ are determined by the subgroup lattice ☎ of G ; they are, essentially, the longest ✽✾✄ G (G4) For any J I , there exists k ★ J such that I chain and the largest Boolean lattice, respectively, ✻ is a maximal subgroup of G J . G J which can be embedded in ✽✾✄ G ☎ . These measures ✸✺✹ k can easily be generalised: we can ask about This means that the group G J acts primitively on the embedding other posets in the subgroup lattice of G . varieties of at least one type in the residue of the One natural measure which springs to mind is the standard flag of type J . size of the largest antichain in ✽✾✄ G ☎ . The rank of a coset geometry for G is at most µ ☎ , Needless to say, there are many other measures ✭✬✄ G while the rank of an RWPri coset geometry is at most which have been used in different circumstances: the µ ✄ G ☎ . In general, it is not true that equality holds, and number of conjugacy classes (the Monster is a so we have two new measures of the size of a group: remarkably small group in this sense); the degree of the smallest faithful permutation representation, or ✕ the maximum rank of a coset geometry; matrix representation (important if we have to do computation in the group); and so on. ✕ the maximum rank of an RWPri coset geometry. 9 11 Other measures? Coset geometries One can also look at the relation between different measures, or between the measure of a group and a Cameron and Cara showed that all the minimal subgroup or quotient. We have seen examples of 1 give rise to RWPri generating sets for S n of size n both of these: e.g. d ✟✗✄ G l ✄ G ☎ and µ ✭✬✄ G l ✄ G ☎ ; and coset geometries. Hence we conclude: ☎✿✡ ☎✿✡ ☎ for any normal subgroup N of G . l ✄ G l ✄ N l ✄ G ✌ N ☎✚✙ The maximum rank of a coset geometry for S n is A similar relation used by Whiston is 1 . Any geometry which meets this bound is n RWPri, and all such geometries are known. µ ✄ G µ ✄ G ✌ N µ ✄ N ☎✔✡ ☎✚✙ for any normal subgroup N of G . It would be The diagram of the geometry corresponding to an interesting to know when the bound is met. independent generating set of the first type (transpositions corresponding to the edges of a Also, given a measure m , we can define m ☎ to be ✟✠✄ G tree T ) is the line graph of T . For the second type, the maximum of m ✄ H ☎ over all subgroups H of G (as the unique transposition in the set corresponds to an we did to get from d to d ✟ ). If the measure m is not isolated node of the diagram. monotonic, this will give us something new; but m ✟ will then of course be monotonic. 10 12

  4. References L. Babai, On the length of subgroup chains in the symmetric group, Commun. Algebra 14 (1986), 1729–1736. P . J. Cameron and Ph. Cara, Independent generating sets and geometries for symmetric groups, in preparation. P . J. Cameron, R. Solomon and A. Turull, Chains of subgroups in symmetric groups, J. Algebra 127 (1989), 340–352. P . Diaconis and L. Saloff-Coste, Walks on generating sets of groups, Invent. Math. 134 (1998), 251–299. M. R. Jerrum, A compact representation for permutation groups, J. Algorithms 7 (1986), 60–78. A. McIver and P . M. Neumann, Enumerating finite groups, Quart. J. Math. (2) 38 (1987), 473–488. R. Solomon and A. Turull, Chains of subgroups in groups of Lie type, I–III, J. Algebra 132 (1990), 174–184; J. London Math. Soc. (2) 42 (1990), 93–100; ibid. (2) 44 (1991), 437–444. J. Whiston, Maximal independent generating sets of the symmetric group, J. Algebra 232 (2000), 255–268. 13

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