Homomorphisms between restricted genera Jules C. Mba Department of - PowerPoint PPT Presentation

Homomorphisms between restricted genera Jules C. Mba Department of Pure and Applied Mathematics University of Johannesburg South Africa jmba@uj.ac.za August 11, 2017 Jules C. Mba (University of Johannesburg) Groups St Andrews 2017, 05-13 August 2017 August 11, 2017 1 / 24

Introduction Homomorphisms Constructing a map that preserves algebraic structure is a natural exercise when dealing with sets having interesting algebraic structure and presents computations advantages. Jules C. Mba (University of Johannesburg) Groups St Andrews 2017, 05-13 August 2017 August 11, 2017 2 / 24

Introduction Aim of the study We focus on the class X 0 of all finitely generated groups with finite commutator subgroup. Given two such groups G 1 and G 2 for which n 1 and n 2 are relatively prime, we aim at establishing a homomorphism between localization genera of such groups under a given finite group F . Jules C. Mba (University of Johannesburg) Groups St Andrews 2017, 05-13 August 2017 August 11, 2017 3 / 24

Introduction Localization Theory The theory of π -localization of groups, where π is a family of primes, appears to have been first discussed in [10, 9] by Mal’cev and Lazard and many others become interested in the theory, such as Baumslag [1, 2] and Bousfield-Kan [3] [9]M. Lazard, Sur les groupes nilpotents et les anneaux de Lie, (French) Ann. Sci. Ecole Norm. Sup. (3) 71 (1954) 101-190 . [10] A.I. Mal’cev, Nilpotent torsion-free groups, (Russian) Izvestiya Akad. Nauk. SSSR. Ser. Mat. 13 (1949) 201-212 . [1]G. Baumslag, Lecture notes on nilpotent groups, Regional Conference Series in Mathematics , No. 2, American Mathematical Society, Providence, R.I. 1971 . [2]G. Baumslag, Some remarks on nilpotent groups with roots, Proc. Amer. Math. Soc. 12 (1961) 262-267 . [3]A.K. Bousfield and D. M. Kan, Homotopy limits, completions and localizations, Lecture Notes in Mathematics , Vol. 304. Springer-Verlag, Berlin-New York, 1972 . Jules C. Mba (University of Johannesburg) Groups St Andrews 2017, 05-13 August 2017 August 11, 2017 4 / 24

Genus of a group In the 1970s, Hilton and Mislin became interested through their work on the localization of nilpotent spaces, in the localization of nilpotent groups. [12] G. Mislin, Nilpotent groups with finite commutator subgroups, Localization in group theory and homotopy theory, and related topics ( Sympos., Battelle Seattle Res. Center, Seattle, Wash. , 1974), 103-120, Lecture Notes in Math., Vol. 418, Springer, Berlin , 1974. Definition Mislin in [12] defines the genus of a finitely generated nilpotent group G denoted by G ( G ), to be the set of all isomorphism classes of finitely generated nilpotent groups H such that G p ∼ = H p for every prime number p . Hilton and Mislin in [7] defined an abelian group structure on the genus set G ( G ) of a finitely generated nilpotent group G with finite commutator subgroup. [7] P. Hilton and G. Mislin, On the genus of a nilpotent group with finite commutator subgroup, Math. Z. 146 (1976), no. 3, 201-211. Jules C. Mba (University of Johannesburg) Groups St Andrews 2017, 05-13 August 2017 August 11, 2017 5 / 24

Non-cancellation set of a group Definition For a finitely generated group G with finite commutator subgroup, the non-cancellation set is the set χ ( G ) of all isomorphism classes of finitely generated group H such that G × Z ∼ = H × Z . The set τ f ( G ) of all isomorphism classes of finitely generated group H such that G π ∼ = H π for every finite set of primes π is called the restricted genus of G . Jules C. Mba (University of Johannesburg) Groups St Andrews 2017, 05-13 August 2017 August 11, 2017 6 / 24

Assigning a natural number n ( G ) to a X 0 -group G Let n 1 be the exponent of T G , let n 2 be the exponent of the group Aut( T G ), and let n 3 be the exponent of the torsion subgroup of the centre of G . Consider n ( G ) = n 1 n 2 n 3 . n = n ( G ) has the property that the subgroup G ( n ) = � g n : g ∈ G � of G belongs to the centre of G and G / G ( n ) is a finite group. Jules C. Mba (University of Johannesburg) Groups St Andrews 2017, 05-13 August 2017 August 11, 2017 7 / 24

Assigning a natural number n ( G ) to a X 0 -group G Let π = { p : p is a prime and p | n ( G ) } . Then the short exact sequence 1 → G ( n ) → G → G / G ( n ) → 1 determines G as an extension of a π ′ -torsion-free finitely generated abelian group G ( n ) by a π -torsion group G / G ( n ) . From [ ? , Proposition 3.1], it follows that the π -localization homomorphism G → G π is injective. Jules C. Mba (University of Johannesburg) Groups St Andrews 2017, 05-13 August 2017 August 11, 2017 8 / 24

Group structure on the noncancellation set Witbooi in [16] shows that the non-cancellation set of a X 0 -group G has a group structure and there is an epimorphism ζ : Z ∗ n / ± 1 → χ ( G ) , where n = n ( G ). [16] P.J. Witbooi, Generalizing the Hilton-Mislin genus group, J. Algebra 239 (2001), no. 1, 327-339. Jules C. Mba (University of Johannesburg) Groups St Andrews 2017, 05-13 August 2017 August 11, 2017 9 / 24

Group structure on the noncancellation set 1 For a nilpotent X 0 -group G , Warfield in [15] shows that χ ( G ) ∼ = G ( G ) . 2 O’Sullivan in [13] shows that for a X 0 -group G , χ ( G ) ∼ = τ f ( G ) . [13] N. O’Sullivan, Genus and cancellation, Comm. Algebra 28 (2000), no. 7, 3387-3400. [15] R. Warfield, Genus and cancellation for groups with finite commutator subgroup, J. Pure Appl. Algebra 6 (1975) 125-132. Jules C. Mba (University of Johannesburg) Groups St Andrews 2017, 05-13 August 2017 August 11, 2017 10 / 24

Homomorphisms between non-cancellation groups Existence of homomorphisms For a semidirect product H = Z m ⋊ ω Z , the authors in [5] showed that there is a well-defined surjective homomorphism Γ : χ ( H ) → χ ( H r ) given by [ K ] → [ K × H r − 1 ] where K is a group such that K × Z ∼ = H × Z and r is a natural number. Thus, in order to compute the group χ ( H r ) one needs only to compute the kernel of the homomorphism Γ. [5]A. Fransman and P. Witbooi, Non-cancellation sets of direct powers of certain metacyclic groups, Kyungpook Math. J. 41 (2001), no. 2, 191-197 . Jules C. Mba (University of Johannesburg) Groups St Andrews 2017, 05-13 August 2017 August 11, 2017 11 / 24

Computation of χ ( G 1 × G 2 ) [16] P.J. Witbooi, Generalizing the Hilton-Mislin genus group, J. Algebra 239 (2001), no. 1, 327-339. Description Witbooi in [16] notice that for any X 0 -groups G 1 and G 2 and for groups K belonging to χ ( G 1 ), the rule K �→ K × G 2 induces a well-defined function θ : χ ( G 1 ) → χ ( G 1 × G 2 ) which is an epimorphism. Jules C. Mba (University of Johannesburg) Groups St Andrews 2017, 05-13 August 2017 August 11, 2017 12 / 24

Category of X 0 -groups under a finite group F Let us fix a finite group F . Let Grp F be the category of groups under F . Here we mean that the objects of Grp F are group homomorphisms ϕ : F → G . Given another object ϕ 1 : F → G 1 , a morphism in Grp F corresponds to a group homomorphism α : G → G 1 such that α ◦ ϕ = ϕ 1 . For a set of primes π , the π -localization of an object ϕ : F → G will be the object ϕ π : F → G π . Then localization is an endofunctor of Grp F . Jules C. Mba (University of Johannesburg) Groups St Andrews 2017, 05-13 August 2017 August 11, 2017 13 / 24

Category of X 0 -groups under a finite group F Let X F be the full subcategory of X 0 -groups under F . We can define the restricted genus Γ f ( ϕ ) = { [ ψ ] | ψ π is isomophic to ϕ π } If F is the trivial group, then X F can be identified with the class X 0 of groups. In line with [16] and in analogy with X 0 -groups we shall write Γ f ( φ ) = χ ( G , φ ). [16] P.J. Witbooi, Generalizing the Hilton-Mislin genus group, J. Algebra 239 (2001), no. 1, 327-339. Jules C. Mba (University of Johannesburg) Groups St Andrews 2017, 05-13 August 2017 August 11, 2017 14 / 24

Category of X 0 -groups under a finite group F Theorem [11, Theorem 2.3] Let ( L , l ) be an object representing a member of χ ( G , h ). Then there exist a subgroup J of G with [ G : J ] finite and [ G : J ] relatively prime to n , such that in Grp F the object F → J is isomorphic to ( L , l ). [11]J.C. Mba and P.J. Witbooi, Induced morphisms between localization genera of groups, Algebra Colloquium , 21:2 (2014) 285-294 . Jules C. Mba (University of Johannesburg) Groups St Andrews 2017, 05-13 August 2017 August 11, 2017 15 / 24

Homomorphisms between non-cancellation groups Existence of homomorphisms Let F be a finite group and consider the homomorphism h : F → G . In [11], a group structure is defined on χ ( G , h ) and an epimorphism ζ : ( Z / n ) ∗ / ± 1 → χ ( G , h ) is established. It is also shown that there exist natural epimorphisms χ ( G , h ) → χ ( G / h ( F )) and χ ( G , h ) → χ ( G , h ◦ i )) . [11]J.C. Mba and P.J. Witbooi, Induced morphisms between localization genera of groups, Algebra Colloquium , 21:2 (2014) 285-294 . Jules C. Mba (University of Johannesburg) Groups St Andrews 2017, 05-13 August 2017 August 11, 2017 16 / 24