R. Kwashira 1, 2 Work with J.C. Mba 3 ( 1 Birmingham, (CoE-MaSS), South Africa, 3 University of Johannesburg, South Africa

Transcription

1 Groups, St Andrews 2017 Non-cancellation group of a direct product R. Kwashira 1, 2 Work with J.C. Mba 3 1 University of the Witwatersrand, South Africa, 2 DST-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS), South Africa, 3 University of Johannesburg, South Africa Birmingham, UK August, 2017 University of the Witwatersrand, South Africa, 2 UK August, /

2 Outline of Talk Introduction Non cancellation set of χ o -groups Non cancellation set of K-groups The category Grp F Non cancelation group of a direct product University of the Witwatersrand, South Africa, 2 UK August, /

3 ABSTRACT Abstract Consider the semidirect product G i = Z ni ωi Z. Methods for computation of the non-cancellation groups χ(g 1 G 2 ) and χ(g k i ), k N were developed in literature. We develop a general method of computing χ(g 1 G 2, h), where h : F G 1 G 1 G 2 and F a nite group. University of the Witwatersrand, South Africa, 2 UK August, /

4 Introduction N f.g. nilpotent group G(N): isomorphism classes of f.g nilpotent groups M such that for every prime p M p N p G(N) has a group structure χ o : class of all G f.g. groups with [G, G] nite Non cancellation set χ(g): set of all isomorphism classes of groups H such that G Z = H Z N is a f.g nilpotent group [N, N] nite then G(N) = χ(n) χ(g) has a group structure similar to the group structure on the Mislin genus of a nilpotent χ o - group H Z = G Z i. H π = Gπ (π for every nite set of primes π χ o - group H, χ(h) coincides with the restricted genus Γ f (H) of H University of the Witwatersrand, South Africa, 2 UK August, /

5 Introduction 2 continued... Fix F nite grp. h : F G be a monomorphism Grp F : category of groups under F objects: (G, h), (G 1, k) group homomorphisms h : F G and k : F G 1 morphisms: group homomorphism β : G G 1 such that β h = k π-localization of an object: h : F G is the object h π : F G π where π is set of primes X F : full subcategory of χ o - groups Γ f (h): restricted genus: isomorphism classes k such that k π is isomorphic to h π for k X F F is a trivial group then X F is identied with χ o -groups F trivial : χ(g, h), non cancellation group of G under F coincides with χ(g) University of the Witwatersrand, South Africa, 2 UK August, /

6 Introduction 3 continued... K be the class of groups: T ω F, F nite rank free abelian group T nite abelian group H = G(m, u) = Z m ν Z, (m, u) = 1. H is a K- group K F full subcategory of Grp F University of the Witwatersrand, South Africa, 2 UK August, /

7 Problem Statement Let G i = Z mi ωi Z and let h : F G 1 G 2 where F is a nite group and h is a monomorphism. We develop a general method for computing χ(g 1 G 2, h). University of the Witwatersrand, South Africa, 2 UK August, /

8 Non cancellation set of χ o -groups χ o - group G : assign a natural number n(g) = n 1 n 2 n 3 n 1 : exponent of T G n 2 : exponent of Aut(T G ) n 3 : exponent of the torsion of the center T ZG G χ o -group with [G : H] <, ; T G = T H W = {H < G, ([G : H], n) = 1} then H Z = G Z. K group, K Z = G Z,then exists J W with J = K. Given J 1, J 2 W, if [G : J 1 ] ±1[G : J 2 ] mod n, then J 1 = J2 non-cancellation set χ(g) coincides with the (nite) set {[J] : J W }. Z n/{1, 1} χ(g) induces a group structure on χ(g) G nilpotent, χ(g) coincides with Hilton-Mislin Genus. University of the Witwatersrand, South Africa, 2 UK August, /

9 Non cancellation set of K-groups G = Z m ω Z ω : Z Aut(Z m ) automorphism of Z m dened ω(1)(t) = ut d : multiplicative order of u modulo m χ(g) = Z /{1, 1} d University of the Witwatersrand, South Africa, 2 UK August, /

10 Non cancellation group of a morphism χ(g, h) n = n(g), X (n) = {u N : (u, n) = 1} Y (G, h) = {u X (n), ; exist K < G, ; [G : K] = u, (K, h K ) member χ(g, h)} G u < G, ; T G G u and [G : G u ] = u for each u Y (G, h) h u : x h(x) induced homomorphism ς : Y (G, h) χ(g, h) Y (G, h) :image of Y (G, h) in Z n, Y (G, h) < Z n G χ o -group, h : F G monomorphism, F nite group ς : Y (G, h)/ ± 1 χ(g, h) induces a group structure on χ(g, h) F trivial, χ(g, h) = χ(g) University of the Witwatersrand, South Africa, 2 UK August, /

11 Non cancellation group of a morphism of an object in K F G = Z m ω Z t = d(g) : smallest invariant factors of Im ω V (G, h) = {u X (t)} s.t K < G, [G : K] = u and (K, h K ) is a member of χ(g, h) V (G, h) be the image of V (G, h) in Z t Choose K < G of G s.t. (K, h K ) member in χ(g, h) and [G : K] = u t n(g) University of the Witwatersrand, South Africa, 2 UK August, /

12 Proposition Proposition Let n = n(g) and ρ : Y (G, h) χ(g, h) be the epimorphism that takes a residue mod n and reduces it mod t. The epimorphism ζ : Y (G, h) χ(g, h) factorises through the epimorphism ξ : ζ Y (G, h) χ(g, h) ρ ξ V (G, h) University of the Witwatersrand, South Africa, 2 UK August, /

13 Theorem Theorem For m V (G, h), the following conditions are equivalent: (a) m ker[v (G, h) χ(g, h)]. (b) There exists α Aut (T ) with v h = h such that α N Aut T Im ω and for an automorphism : Im ω Im ω dened by v α vα 1, we have det( ) = ±m 1 V (G, h). University of the Witwatersrand, South Africa, 2 UK August, /

14 Non cancelation group of (G1 G2, h) G i = G(m i, u i ) with gcd(m 1, m 2 ) = 1, gcd(m i, u j ) = 1, i, j = 1, 2 d i : multiplicative order of u i modulo m i d = lcm(d 1, d 2 ) and m = lcm(m 1, m 2 ). If gcd(d 1, d 2 ) = 1 let t = d 1 d 2 and if gcd(d 1, d 2 ) 1 let t = gcd(d 1, d 2 ) G 1 G 2 = (Z m 1 Z m2 ) ω Z 2 ω : Z 2 Aut(Z m 1 Z m2 ) ω(ɛ i ) = ω i : (t 1, t 2 ) (u δ(i,1) 1 t 1, u δ(i,2) 2 t 2 ) {ɛ 1, ɛ 2 } standard basis of Z 2 J = Im ω =< ω 1, ω 2 > is a free Z d -module University of the Witwatersrand, South Africa, 2 UK August, /

15 Non cancelation group of (G1 G2, h) continued... σ End(J), det (σ) Z d J = N Aut(TG 1 G 2 ) J α Aut(T G 1 G2 ), α inner automorphism of J: α q 2 be a multiple of q such that q 2 q divides m. : v α v α 1 e 1 = (1, 0), e 2 = (0, 1) :elements of T G 1 G2 F = {aq 2 e 2 : a Z} F < T G 1 G2 h : F G 1 G 2 inclusion map University of the Witwatersrand, South Africa, 2 UK August, /

16 An inner automorphism of Aut(T G1 G 2 ) Fix α J such that α(x) = x x F exists a 2 2 matrix (α ij ) of integers such that α(e i ) = 2 j=1 α jie j Suppose that α is the inner automorphism of J determined by α. Proposition For the matrix (α ij ), α ii is a unit modulo m for i = 1, 2. University of the Witwatersrand, South Africa, 2 UK August, /

17 Proof α(0, q 2 ) = (0, q 2 ) since (0, q 2 ) F α(0, q 2 ) = (q 2 α 12, q 2 α 22 ) = (0, q 2 ) m divides α 12 q divides α 12, while α 22 is a unit modulo m ( ) α11 tq [α] = u α 21 det(α) = α 11 u α 21 tq Claim: α 11 is a unit modulo m α 11 is not a unit modulo m let p common prime divisor of α 11 and m p divides q and p divides det(α) contradiction!!! ( det(α) is a unit modulo m ) (α is an automorphism). Thus α 11 is a unit modulo m. University of the Witwatersrand, South Africa, 2 UK August, /

18 Proposition B Proposition The inner automorphism α of J. of J coincides with the identity automorphism University of the Witwatersrand, South Africa, 2 UK August, /

19 Proof for α exists a matrix ( ) ij of integers such that 1i ω i = ω each i let ω i = v i. Also ω i = αω i α 1 = v i αω i (e i ) = α(u i e i ) = 2 j=1 uδ(i,j) α i ji e j ( 2 ) v i α(e i ) = v i j=1 α jie j = 2 2 j=1 uδ(i,j) α i ji e j = 2 j = i α ii ii u u i i mod m is a unit modulo m ii 1 mod d ii 1 mod t, (t d) j=1 α jiu δ(i,j) ji e i j j=1 α jiu δ(i,j) ji e i j αω i = v i α 1 ω 2i 2 for University of the Witwatersrand, South Africa, 2 UK August, /

20 Proof continued... j i αω i (e j ) = α(u δ(i,j) (e j )) = α(e j ) = 2 v i α(e j ) = 2 α k=1 kj uk ki e k k=1 α kj e k Therefore 2 α k=1 kj e k = 2 α ki k=1 kj u e k k k = j. Since α jj is a unit modulo m ji then u 1 mod m, that is, j ji = 0 mod d Consequently 0 mod t. Thus det ( ji α ) = 1 is identity on J α University of the Witwatersrand, South Africa, 2 UK August, /

21 Proposition C Proposition The pair (G 1 G 2, h) is an object of K F for which (G 1 G 2, h) = Z t / ± 1. Proof Proposition follows from Theorem 1 and Proposition 3 University of the Witwatersrand, South Africa, 2 UK August, /

22 Bibliography A. Fransman and P. Witbooi, Non-cancellation sets of direct powers of certain metacyclic groups, Kyungpook Math. J. 41 (2001), no. 2, P. Hilton and G. Mislin, On the genus of a nilpotent group with nite commutator subgroup, Math. Z., 146 (1976), J. C. Mba and P. J. Witbooi, Induced morphisms between localization genera of groups, Algebra Colloquium, 21: 2 (2014) G. Mislin, Nilpotent groups with nite commutator subgroups, in: P. HILTON (ED), Localization in Group Theory and Homotopy Theory, Lecture Notes in Mathematics, 418 (1974), , Springer-Verlag Berlin. N. O'Sullivan, Genus and Cancellation, Comm. Algebra 28 (7) (2000), University of the Witwatersrand, South Africa, 2 UK August, /

23 Bibliography continued... D. Scevenels and P. Witbooi, Non-cancellation and Mislin genus of certain groups and H o spaces, J. Pure Appl. Algebra, 170 (2002), R. Wareld, Genus and cancellation for groups with nite commutator subgroup, J. Pure Appl. Algebra 6 (1975), P. Witbooi, Generalizing the Hilton-Mislin genus group, J. Algebra 239 (1) (2001), P. J. Witbooi, Non-cancellation for groups with non-abelian torsion, J. Group Theory 6 (2003), no. 4, University of the Witwatersrand, South Africa, 2 UK August, /