ISCHIA GROUP THEORY 2016, March 29th–April 2nd Groups of rational interval exchange transformations Agnieszka Bier and Vitaliy Sushchanskyy Institute of Mathematics, Silesian University of Technology, Poland [email protected] Interval exchange transformations Rational interval exchange transformations Subgroups of RIET defined by supernatural numbers Let I denote the half-open interval [0 , 1) and let An interval exchange transformation f = ( π, σ ) ∈ IET defined by the rational partition π = {a0, a 1, ..., a n} where all ai are rational numbers is called rational iet . The subset In particular, if f = ( πni, σ ) ∈ RIET( ni) and ni+1 = k · ni then the diagonal embedding π = {a0, a 1, ..., a n}, 0 = a0 ≤ ai < a i+1 ≤ an = 1 of all rational iets is a subgroup of IET, which we denote by RIET. ϕi is defined by the following rule: Every rational interval exchange transformation may be considered as a transformation be a partition of I into n subintervals [ a , a ), i = 0 , ..., n − 1. A left-continuous l · n + j l · n + j + 1 l · n + σ(j) l · n + σ(j) + 1 i i+1 defined by the partition π of interval I into n subintervals of equal length. f ϕi i , i = i , i , bijection f : [0 , 1) → [0 , 1) acting as piecewise translation, i.e. shuffling the subintervals n pi ni+1 ni+1 ni+1 ni+1 If f = ( π, σ ), where π = {0, a , ..., a n , 1}, where ai = ∈ Q, is a ratio- [ai, a i+1 ), is called an interval exchange transformation (iet) of I. The group of all 1 −1 qi interval exchange transformations of I is denoted by IET. nal interval exchange transformation, then there exists a rational partition πq into for all j = 0 , 1, ..., n i − 1 and l = 0 , 1, ..., k − 1. q = LCM (q1, ..., q n−1) subintervals of equal length such that f shuffles the q subintervals The groups RIET( n) together with the diagonal embeddings form a direct system of ′ ′ ′ The action of an exemplary iet for n = 6 is demonstrated on the scheme below: defined by πq according to a permutation σ ∈ Sq such that the action of f = ( πq, σ ) groups. The corresponding direct limit: 0 1 on I is equivalent to the action of f. Moreover, if f and g are two rational iets, then there exists a partition πm of the interval I RIET(¯n) = lim RIET( ni) i into m equally sized subintervals, such that f and g act on I by shuffling the subintervals defined by πm (it is enough to take m as the least common multiplier of all endpoints of is a subgroup of RIET. partitions defining f and g). ˆ ∞ For instance, if M = pi is the characteristic of the divisible sequencem ¯ , then piQ∈P Remark. RIET is a dense subgroup of IET. RIET(m ¯ ) = RIET. 0 1 Every transformation f ∈ IET may be represented in a (unique) canonical form as a Supernatural numbers pair Results f = ( π, σ ), A sequence of natural numbersn ¯ = ( n1, n 2, ... ) is called the divisible sequence , if ni|ni+1 Letn ˆ be a supernatural number with characteristicn ¯ . Then where π is a partition of I into n subintervals ( n being the smallest possible), and σ ∈ Sn (ni divides ni+1 ) for every i ∈ N. In the divisible sequencen ¯ the set of prime divisors of ni is a permutation of the set of n elements. is contained in the set of prime divisors of ni+1 and this includes also the multiplicities of • Ifn ˆ is infinite, then the subgroup RIET(¯n) is isomorphic to the homogeneous symmetric these divisors. Thus with every divisible sequencen ¯ one may associate the supernatural group Sn¯. number nˆ, defined as the formal product • Ifn ˆ is infinite, then the subgroup RIET(¯n) is dense in RIET. εi Topology on IET nˆ = pi , pYi∈P • RIET(ˆn) is either finite or locally finite. In particular RIET(ˆn) is finitely generated if and only if it is finite. Let IET σ denote the set of all iets with canonical form defined with a given permutation where P denotes the (naturally ordered) set of all primes, and εi ∈ N ∪ { 0, ∞} for every σ ∈ Sn. The natural topology on IET arises from the mutual correspondence between i ∈ N. • Ifn ¯ = ( n1, n 2, ... ) andm ¯ = ( ni, n i+1 , ... ), i > 1, then RIET(¯n) = RIET(m ¯ ). the subsets IET σ, σ ∈ Sn+1 , and the standard n-dimensional open simplex The supernatural numbern ˆ associated to the divisible sequencen ¯ is called the charac- n+1 teristic of this sequence. • For every prime p the subgroup RIET( p∞) is the minimal dense subgroup of IET in n+1 ∆n = {(x1, x 2, ..., x n+1 ) ∈ R+ | xi = 1 }, the lattice of all subgroups of RIET defined by supernatural numbers. Xi=1 • If 2 ∞|nˆ then the group RIET(ˆn) is perfect, i.e. RIET(ˆn)′ = RIET(ˆn). which is given by: Subgroups of RIET defined by supernatural numbers ∞ ′ ψ : IET σ −→ ∆n • If 2 ∤ nˆ then the derived subgroup RIET(ˆn) is a proper subgroup of RIET(ˆn) and ψ({a0, a 1, ..., a n+1 }, σ ) = ( a1 − a0, a 2 − a1, ..., a n+1 − an). consists of all the iets from RIET(ˆn), which are defined by even permutations. By RIET( n) = {f ∈ RIET | f = ( πn, σ ), σ ∈ Sn}, n ∈ N we denote the subgroup of RIET, isomorphic to S . For a divisible sequencen ¯ = ( n , n , ... ) we define the diagonal The assumptions imply that IET is a disjoint union of all IET σ, and hence if all of IET σ n 1 2 • RIET(2 ∞) is generated by the set S of all rational iets defined as: are defined to be open, we obtain the topology on IET. embeddings ϕi : RIET( ni) ֒→ RIET( ni+1 ), n−1 We note that IET is not a topological group with this topology, as the operation of S = {(π2n, σ ) | σ = ( i, i + 1) , i ≤ 2 } composition of iets is not continuous. where ni|ni+1 , which correspond to the diagonal embeddings of Sni into Sni+1 . References [1] Bier A., Sushchanskyy V., ,,Dense subgroups in the group of interval exchange transformations”, Algebra Discrete Math. 17 (2014), p. 232 – 247 [2] Bier A., Oliynyk B., Sushchanskyy V., ,,Generating sets of RIET”, to appear Non-Soluble and Non-p-Soluble Length of Finite Groups Bounding the non-p-Soluble length of finite group with condi- tions on their Sylow p-subgroups Yerko Contreras Rojas PhD student UnB Universidade de Bras´ılia. [email protected] Abstract Results Let p be a prime. Every finite group G has a normal series each of whose quotients either is p-soluble or is a direct product of nonabelian simple groups of orders divisible by p. The non-p- In the sequel we give some useful definitions and we present some positive answer soluble length λp(G) is defined as the number of non-p-soluble quotients in a shortest series of this kind. to Problem A. For instance in [5] a positive answer was obtained in the case of any We deal with the question whether, for a given prime p and a given proper group variety V, variety that is a product of varieties that are either soluble or of finite exponent. the non-p-soluble length λp(G) of a finite group G whose Sylow p-subgroups belong to V is Now we define a group variety, that contains the varieties of soluble and of finite bounded. exponent groups like particular cases, and offer us some groups varieties for which In joint work with Pavel Shumyatsky, we answer the question in the affirmative in some cases (working separately the case p =2) for varieties of groups in which the commutators have some Problem A has positive answer. restrictions about their order. Definition. Let W(w,e) be the variety of all groups in which we-values are trivial, where w is a group-word and e is a positive integer. Non-Soluble and Non-p-Soluble Length We obtain the following theorem [2]: Every finite group G has a normal series each of whose quotient either is soluble Theorem 1. Let k,e be positive integers and p an odd prime. Let P be or is a direct product of nonabelian simple groups. In [5] the nonsoluble length of a Sylow p-subgroup of a finite group G and assume that P belongs to e G, denoted by λ(G), was defined as the minimal number of nonsoluble factors in a W(δk,p ). Then λp(G) ≤ k + e − 1. series of this kind: if Using the previous theorem and some others tools we obtain a generalization in the 1= G0 ≤ G1 ≤···≤ G2h+1 = G odd case of the result of Shumyatsky and Khukhro about the product of varieties is a shortest normal series in which for i even the quotient Gi+1/Gi is soluble (pos- sibly trivial), and for i odd the quotient Gi+1/Gi is a (non-empty) direct product of Theorem 2. Let G be a finite group of order divisible by p, where p nonabelian simple groups, then the nonsoluble length λ(G) is equal to h. For any is an odd prime. If a Sylow p-subgroup P of G belongs to the vari- prime p, a similar notion of non-p-soluble length λp(G) was defined by replacing W W W ety (δk1,e1) (δk2,e2) ··· (δkn,en), then λp(G) is {k1,e1,...,kn,en}- “soluble” by “p-soluble” and “simple” by “simple of order divisible by p”.