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Finite Groups 4 3. Profinit INTRODUCTION TO PROFINITE GROUPS MIMAR SINAN FINE ARTS UNIVERSITY (ISTANBUL) 30.1.2012 WOLFGANG HERFORT Dedicated to Peter Plaumann Abstract. A profinite space / group is the projective limit of finite sets / groups. Galois theory offers a natural frame in order to describe Galois groups as profinite groups. Profinite groups have properties that correspond to some of finite groups: e.g., each profinite group does have p-Sylow subgroups for any prime p. In the same vein, every pro-solvable group (the projective limit of an inverse system of finite solvable groups) has Hall subgroups for any given set of primes. Any group can be equipped with the profinite topology turning it into a topological group. A basis of neighbourhoods of the identity-element consists of all normal subgroups of finite index. Any such group allows a completion w.r.t. this topology { the profinite completion.A free profinite group is the profinite completion of a free group. This can be considered an instance of the amalgamated free product and of the HNN extension (Higman-Neumann- Neumann). I do not include cohomological topics in this note. Contents 1. Projective limits1 2. Profinite groups are \large" finite groups4 3. Profinite topology6 4. Free constructions8 5. Acknowledgements 11 References 11 6. Logfile 11 1. Projective limits Definition 1. Let (I; ≤) denote a set with a partial order order `≤', directed up- wards. So, `≤' is a reflexive, antisymmetric and transitive relation on I and for all indices i; j 2 I there is k, so that i ≤ k and j ≤ k. For every i 2 I let Xi be a set; we assume that for all i ≤ j there is a map φji so that for all Indices k ≤ j ≤ i the diagram φij X X i RR / j RRR llll φik R( vllφjk Xk commutes. 1991 Mathematics Subject Classification. Primary 20E06; Secondary 20E18, 20E08. Cordial thanks to the Department of Mathematics of Mimar Sinan University for the generous invitation and to the Außeninstut der Technischen Universit¨atWien for travelling funding. 1 2 WOLFGANG HERFORT Here is an example: ? 6 oy ::: (1; 2; 3; 4; 5; 6; 6;:::) 5 o 5 o ::: (1; 2; 3; 4; 5; 5;:::) 4 o 4 o 4 o ::: (1; 2; 3; 4; 4;:::) 3 o 3 o 3 o 3 o ::: (1; 2; 3; 3;:::) 2 o 2 o 2 o 2 o 2 o ::: (1; 2; 2;:::) 1 o 1 o 1 o 1 o 1 o 1 o ::: (1; 1;:::) In this example I is the set N of natural numbers and `≤' is the natural ordering on N. For i 2 N set Xi := f1; 2; : : : ; ig and let the arrows indicate the maps φi+1i. E.g., φ43(4) = 3, φ43(i) = 3 for i ≥ 3 and φ43(i) = i for i < 4. Definition 2. In order to define the projective limit of the system (I; ≤), proceed as follows: Q (1) Form the Tychonov product P := Xi (which consists of all the maps S i2I f : I ! i2I Xi with f(i) 2 Xi). (2) The projective limit X := lim X is the subset of all those f 2 P satis- −i2I i fying the compatibility relations φij(f(i)) = f(j) for all j ≤ i. (3) The φi : P ! Xi satisfy φijφi = φj for all indices j ≤ i and are nothing but the canonical projections of the Tychonov product restrichted to X. Theorem 3. The following statements for a set X are equivalent: • X is a compact topological space; moreover it is totally disconnected and Hausdorff. • X is the projective limit of an inverse system of finite sets. Q • X is a closed subset of the cartesian product j2J Xj of finite discrete topological spaces Xj. Such X is a profinite space. In the above example Xi = f1; 2; : : : ; ig. The elements of X are the infinite \rays" (fi), for which 1 ≤ f(i) ≤ i. In particular we find \rays" of the sort (1; 2; 3; 4; : : : ; i; i; i; i; : : :) for i 2 N and the special one 1 := (1; 2; 3; 4; 5; 6;:::). Thus lim Xi coincides with the Aleksandrov-compactification of the natural −i2N numbers N [ f1g. The projective limit lim X of an inverse system (X ; φ ) enjoys a universal −i i i ij property: whenever Z is a profinite space and there are continuous maps i : Z ! Xi so that for all indices i ≤ j the diagrams Z i !9! # lim X φi ! −k k / Xi GG GG GG φij j φj GG G# . Xj commute then there is a universal map !, so that φi! = i holds for all i 2 I. INTRODUCTION TO PROFINITE GROUPS 3 Example 4. Consider the diagram 0000 ::: ggg 000 osgg 0001 ::: iii 00 tiii 0010 ::: jUUU g | UUU sgggg || 001 o 0011 ::: || || 0 ~ `BB BB BB 010 o 0100 ::: B ii kWWWW tiiii W 01 jU 0101 ::: UUUUU 011 o 0110 ::: kWWWWW 0111 ::: This gives rise to defining an inverse system. How to define formally φij? Example 5. Let X = lim X and no X be the empty set. Show the following −i i i assertions: (1) X is not the empty set. (2) For every point x 2 X and every neighbourhood U of x there is i 2 I so that −1 φi (φi(x)) is a closed and open (a clopen) neighbourhood of x contained in U. ¯ T −1 (3) For a subset A ⊆ X its topological closure A equals i2I φi φi(A). When ij denotes the restriction of φij to φi(A) then (φi(Ai); ij) is an inverse system. Show that A = lim φ (A ). −i i i Example 6. (Vietoris-topology) For a profinite space X = lim X denote by C(X) −i i the set of all closed non empty subsets of X. For a continuous map f : X ! Y define C(f): C(X) ! C(Y ) by C(f)(A) := ff(a) j a 2 Ag. The map C(−) is a functor. We obtain the inverse system (C(X );C(φ )). Show that C(X) = lim C(X ). i ij −i i Show that the subsets W (U1;:::;Un) of C(X), where Ui are clopen subsets of X, and n [ W (U1;:::;Un) := fC 2 C(X) j C ⊆ Uj ^ C \ Cj 6= ;; j = 1; : : : ng; j=1 form a basis of the Vietoris-topology. Example 7. Let k be a field and I the set of all finite Galois extensions of k. The relation K ≤ L if k ⊆ K ⊆ L is a partial ordner. It is well-known that for all finite Galois extensions K and L there is a finite Galois extension M so that K ≤ M and L ≤ M. Hence ≤ is a directed order. Lemma 8. If for all i 2 I the set Xi is a group and all φji are homomorphisms then one can define a group multiplication on X so that X becomes a topological group and the canonical projections become continuous homomorphisms. Q Proof. The product P := i2I Xi is a group with group operation (xi)(yi) := (xiyi). Observe the continuity of this operation w.r.t. the product topology. The projective limit lim X is a closed subgroup of P . −i i Example 9. As in Ex.7 consider the finite Galois extensions of a field k. Let G(KjL) be the Galois group of a Galois extension K of L. Galois' theorem says 4 WOLFGANG HERFORT that G(Ljk) is isomorphic to G(Kjk)=G(KjL). K O O G(KjL) L G(Kjk) O O G(Ljk) =∼ G(Kjk)=G(KjL) k Hence there is K ≤ L (L ⊆ K) and a canonical homomorphism φKL : G(Kjk) ! G(Ljk). The inverse system (G(Kjk); φ ) gives rise to the projective limit lim G(KjL). LK −K ^ S ^ Let k := K K. It is clear that k is a field and an algebraic and normal extension of k. The Galois group G(k^jk) agrees with the projective limit lim G(KjL). −K Replacing in the diagram K by k^ one observes that G(k^jL) is a normal sub- group of finite index in G(k^jk). The Krull-topology on G(k^jk) has a basis of open neighbourhoods of the identity consisting of all subgroups of the form G(k^jL). Example 10. For a profinite group G consider the set N of all open normal subgroup and the order `' given as N ≤ M whenever M is a subgroup of N. Show that (1) every N 2 N is closed. (2)( N ; ) is a directed set. (3) Let φN : G ! G=N denote the canonical homomorphism. There exist induced homomorphisms φNM : G=M ! G=N whenever M is contained in N. Obtain that G = lim G=N. −N Example 11. Show that for a profinite group G the closed subgroups form a closed subset S(G) of C(G). For an open normal subgroup N and a finite subset Y of G set W (N) := fS 2 S(G)jSN = hY iNg. Show that the sets W (Y; N) form a basis of the induced Vietoris-topology on S(G). 2. Profinite groups are \large" finite groups A projective limit of an inverse system (Gi; φij) of finite p-groups (p prime) is a pro-p group. A profinite group G is pro-p if and only if G=N is a finite p-group for every open normal subgroup N of G. Example 12. Let us return to Ex.4. Every natural number n has a binary expansion 0 1 2 k n = a02 + a12 + a22 + ::: + ak2 with aj 2 f0; 1g.
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