By Michael Huber

On account of the category of the finite basic teams, it hasbeen attainable lately to symbolize Steiner t-designs, that's t -(v, ok, 1) designs,mainly for t = 2, admitting teams of automorphisms with sufficiently strongsymmetry houses. although, regardless of the finite basic workforce category, forSteiner t-designs with t > 2 each one of these characterizations have remained longstandingchallenging difficulties. specifically, the decision 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 guide of IncidenceGeometry, Elsevier technological know-how, Amsterdam, 1995, p. 273), yet possibly datingback to 1965).The current paper keeps the author's paintings (see Huber (J. Comb. concept 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 end up furthermorethat there are not any non-trivial flag-transitive Steiner 6-designs. either effects depend on theclassification of the finite 3-homogeneous permutation teams. furthermore, we surveysome of the main basic effects on hugely symmetric Steiner t-designs.

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2 (Comparison Test). Let an , ries with an ≤ bn for all n ≥ 1. If bn < ∞, then then bn = ∞. bn be nonnegative sean < ∞. 6 Nonnegative Series and Decimal Expansions 33 Stated this way, the theorem follows from the ordering property for sequences and looks too simple to be of any serious use. ⊓ ⊔ In fact, we use it to express every real as a sequence of naturals. 3. Let b = 9 + 1. 3), then ∞ dn b−n n=1 sums to a real x, 0 ≤ x ≤ 1. Conversely, if 0 ≤ x ≤ 1, there is a sequence of digits d1 , d2 , · · · , such that the series sums to x.

Now we define the limit of an arbitrary sequence. Let (an ) be any sequence, and let (a∗n ), (an∗ ), a∗ , a∗ be the upper and lower sequences together with their limits. We call a∗ the upper limit of the sequence (an ) and a∗ the lower limit of the sequence (an ). If they are equal, a∗ = a∗ , we say that L = a∗ = a∗ is the limit of (an ), and we write lim an = L. nր∞ Alternatively, we say an approaches L or an converges to L, and we write an → L as n ր ∞ or just an → L. If they are not equal, a∗ = a∗ , we say that (an ) does not have a limit.

Then (En ) is an error sequence for the Cauchy sequence of partial sums of |An |. Now in the difference Sn − sn , there is cancellation, the only terms remaining being of one of two forms, either Ak = af (k) with f (k) > n or ak with k = f (j) with j > n (this is where surjectivity of f is used). Hence in either case, the absolute values of the remaining terms in Sn − sn are summands in the series en + En , so |Sn − sn | ≤ en + En → 0, as n ր ∞. This completes the derivation of the absolute portion of the theorem.