By Michael Huber

On account of the type of the finite basic teams, it hasbeen attainable lately to signify Steiner t-designs, that's t -(v, okay, 1) designs,mainly for t = 2, admitting teams of automorphisms with sufficiently strongsymmetry houses. although, regardless of the finite basic workforce type, forSteiner t-designs with t > 2 every one of these characterizations have remained longstandingchallenging difficulties. particularly, the selection of all flag-transitiveSteiner t-designs with three ≤ t ≤ 6 is of specific curiosity and has been open for about40 years (cf. Delandtsheer (Geom. Dedicata forty-one, p. 147, 1992 and instruction manual of IncidenceGeometry, Elsevier technology, Amsterdam, 1995, p. 273), yet possibly datingback to 1965).The current paper maintains the author's paintings (see Huber (J. Comb. idea Ser.A ninety four, 180-190, 2001; Adv. Geom. five, 195-221, 2005; J. Algebr. Comb., 2007, toappear)) of classifying all flag-transitive Steiner 3-designs and 4-designs. We provide acomplete type of all flag-transitive Steiner 5-designs and turn out furthermorethat there are not any non-trivial flag-transitive Steiner 6-designs. either effects depend upon theclassification of the finite 3-homogeneous permutation teams. furthermore, we surveysome of the main common effects on hugely symmetric Steiner t-designs.

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4. THE MODULUS EQUATION AND CLASS GROUPS 51 Proof We use Dirichlet’s theorem on primes in arithmetic progession (see [13]). We choose a prime r ≡ 1 (mod 3) with (r − 1)/3 ≥ 8q + 12. By Dirichlet’s theorem, there are inﬁnitely many primes p with ordr (p) = 3. 22, Q(ξrp has an inﬁnite class ﬁeld tower for each of these p. 16. 24 Let m = m p where p is an odd prime with (p, m ) = 1. Let 2b be the exact power of 2 dividing p − 1. Let ε = 0 if −1 is a power of p modulo m and ε = 1 otherwise. Then Q(ξm ) has a complex subﬁeld K of absolute degree 2b−ε ϕ(m )/ordm (p) with d2 ClK ≥ ϕ(m )/(2ε ordm (p)) − 1.

Our bounds on p-ranks are comparable to, but not as general as those obtained in [119] using heavy number theoretic machinery. Before we turn our attention to cyclotomic ﬁelds, we explain the basic idea behind the results of the present section. First we recall some facts on CM -ﬁelds, see [130, p. 38]. , K = K + ( α) where K + is totally + real and α ∈ K is totally negative. , α−1 (α(x)) = β −1 (β(x)) for all x ∈ K and all imbeddings α, β of K in C. b) We have α(x) = α(x) for all x ∈ K and all imbeddings α of K in C.

S } be a set of generators for D. Let T be a set of primes of k with T ∩ T = ∅. Denote the ramiﬁcation index of P ∈ T in K/k by R(P ). , s). 4. 4. THE MODULUS EQUATION AND CLASS GROUPS 45 Moreover, L is contained in the subgroup of ClK generated by the primes of K above primes in T . Proof We ﬁrst need some notation. Write S := T ∪ T . Write R(P ) = R(P ) R(P ) for P ∈ T . We have P = PK for P ∈ S where each PK is an ideal of K. For the convenience of the reader, we give a table of the notations we need for this proof.