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 X0 of all finitely generated groups with finite commutator subgroup. Given two such groups G1 and G2 for which n1 and n2 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 Gp ∼ = Hp 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 τf (G) of all isomorphism classes of finitely generated group H such that Gπ ∼ = Hπ 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 X0-group G

Let n1 be the exponent of TG, let n2 be the exponent of the group Aut(TG), and let n3 be the exponent of the torsion subgroup of the centre of G. Consider n(G) = n1n2n3. n = n(G) has the property that the subgroup G (n) = gn : 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 X0-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 → Gπ 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 X0-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 X0-group G, Warfield in [15] shows that

χ(G) ∼ = G(G) .

2 O’Sullivan in [13] shows that for a X0-group G,

χ(G) ∼ = τf (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 = Zm ⋊ω Z, the authors in [5] showed that there is a well-defined surjective homomorphism Γ : χ(H) → χ(Hr) given by [K] → [K × Hr−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(Hr)

• ne 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(G1 × G2)

[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 X0-groups G1 and G2 and for groups K belonging to χ(G1), the rule K → K × G2 induces a well-defined function θ : χ(G1) → χ(G1 × G2) 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 X0-groups under a finite group F

Let us fix a finite group F. Let GrpF be the category of groups under F. Here we mean that the objects of GrpF are group homomorphisms ϕ : F → G. Given another object ϕ1 : F → G1, a morphism in GrpF corresponds to a group homomorphism α : G → G1 such that α ◦ ϕ = ϕ1. For a set of primes π, the π-localization of an object ϕ : F → G will be the object ϕπ : F → Gπ. Then localization is an endofunctor of GrpF.

Jules C. Mba (University of Johannesburg) Groups St Andrews 2017, 05-13 August 2017 August 11, 2017 13 / 24

Category of X0-groups under a finite group F

Let XF be the full subcategory of X0-groups under F. We can define the restricted genus Γf (ϕ) = {[ψ] | ψπ is isomophic to ϕπ} If F is the trivial group, then XF can be identified with the class X0 of groups. In line with [16] and in analogy with X0-groups we shall write Γf (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 X0-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 GrpF the object F → J is isomorphic to (L, l).

[11]J.C. Mba and P.J. Witbooi, Induced morphisms between localization genera

• f 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

• f 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

Introduction

Non-existence of homomorphisms

In [11], computation methods of χ(G, h) in the special case G is a semidirect product T ⋊ω Zk are used in a very particular example to provide a concrete computation of χ(G, h). It is used to show that there doesn’t exist any homomorphism γ to make the following diagram commutative χ(K, h)

α

• β
• ❃

❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃

χ(K/h(F)) χ(K)

γ

• ④

④ ④ ④ ④ ④ ④ ④ ④ ④ ④ [11]J.C. Mba and P.J. Witbooi, Induced morphisms between localization genera

• f 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 17 / 24

Homomorphisms between localization genera

Notation

Fix any m ∈ N. Let X(m) = {u ∈ N| (u, m) = 1}. Now consider any G ∈ X0 and let n = n(G). Let Y (G, h) be the set of all u ∈ X(n) for which there exists a subgroup J of G with [G : J] = u and such that the

• bject (J, hJ) represents a member of χ(G, h). Here hJ is the induced

homomorphism obtained from h by restriction of the codomain. For each u ∈ Y (G, h), let us choose a subgroup Gu of G such that TG ⊆ Gu and [G : Gu] = u. Let hu : F → Gu be the induced homomorphism defined by hu : x → h(x). Now let us denote the isomorphism class of the object hu of XF by [Gu, hu]. Then we obtain a function ξ : Y (G, h) → χ(G, h). Let Y ∗(G, h) denote the image of Y (G, h) in Z∗

n.

Jules C. Mba (University of Johannesburg) Groups St Andrews 2017, 05-13 August 2017 August 11, 2017 18 / 24

Homomorphisms between localization genera

Theorem [11, Theorem 2.5]

a)

Y ∗(G, h) is a subgroup of Z∗

n.

b)

The function ξ induces a (well-defined) function ζ : Y ∗(G, h)/ ± 1 → χ(G, h).

c)

The fibre ζ−1[G, h] of ζ over [G, h] is a subgroup of Y ∗(G, h)/ ± 1.

d)

For any [K, k] ∈ χ(G, h), ζ−1[K, k] is a coset of ζ−1[G, h].

[11]J.C. Mba and P.J. Witbooi, Induced morphisms between localization genera

• f 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 19 / 24

Homomorphisms between localization genera

Theorem

Let (G1, h1) and (G2, h2) be such that n1 = n(G1) and n2 = n(G2) are relatively prime. Then,

1 There is a homomorphism

α : Y ∗(G1, h1)/ ± 1 → Y ∗(G2, h2)/ ± 1 defined by u → n⌊ln(u)⌋

1

.

2 There are homomorphisms ϕ and β such that the following diagram is

commutative. Y ∗(G1, h1)/ ± 1

ξ1

• β
• ◆

◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆

χ(G1, h1)

ϕ

• χ(G2, h2)

If β is surjective, then ϕ is an epimorphism.

Jules C. Mba (University of Johannesburg) Groups St Andrews 2017, 05-13 August 2017 August 11, 2017 20 / 24

Homomorphisms between localization genera

Corollary

Let (G1, h1) and (G2, h2) be such that n1 = n(G1) and n2 = n(G2) are relatively prime. Let ϕ and β the following homomorphisms Y ∗(G1, h1)/ ± 1

ξ1

• β
• ◆

◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆

χ(G1, h1)

ϕ

• χ(G2, h2)

If β is surjective, then ϕ is an epimorphism.

Jules C. Mba (University of Johannesburg) Groups St Andrews 2017, 05-13 August 2017 August 11, 2017 21 / 24

Homomorphisms between localization genera

Proposition [16, Proposition 6.1]

. Suppose that we have groups A, B and C together with a homomorphism β : A → C and a surjective group homomorphism γ : A → B. If α : B → C is a function (between sets) such that α ◦ γ = β, then α is a

• homomorphism. Moreover, if β is surjective, then α is also surjective.

[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 22 / 24

THANK YOU

Jules C. Mba (University of Johannesburg) Groups St Andrews 2017, 05-13 August 2017 August 11, 2017 23 / 24

• G. Baumslag, Lecture notes on nilpotent groups, Regional Conference

Series in Mathematics, No. 2, American Mathematical Society, Providence, R.I. 1971.

• G. Baumslag, Some remarks on nilpotent groups with roots, Proc.
• Amer. Math. Soc. 12 (1961) 262-267.

A.K. Bousfield and D. M. Kan, Homotopy limits, completions and localizations, Lecture Notes in Mathematics, Vol. 304. Springer-Verlag, Berlin-New York, 1972. [1] C. Casacuberta and P. Hilton, Calculating the Mislin genus for a certain family of nilpotent groups, Comm. Algebra 19 (1991), no. 7, 2051-2069. [2]A. Fransman and P. Witbooi, Non-cancellation sets of direct powers

• f certain metacyclic groups, Kyungpook Math. J. 41 (2001), no. 2,

191-197. [3]P. Hilton and C. Schuck, On the structure of nilpotent groups of a certain type, Topol. Methods Nonlinear Anal. 1 (1993), no. 2, 323-327. [4]P. Hilton and G. Mislin, On the genus of a nilpotent group with

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