H-Transversals in H-Groups

Transcription

1 International Journal of Algebra, Vol. 8, 2014, no. 15, HIKARI Ltd, H-Transversals in H-roups Swapnil Srivastava Department of Mathematics & Statistics SHIATS, Allahabad U.P., India Punish Kumar Department of Mathematics & Statistics SHIATS, Allahabad U.P., India Corresponding author Copyright c 2014 Swapnil Srivastava and Punish Kumar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper, we have defined the concept of H-subgroup and H- transversal in an H-group and then we have shown that there is a canonical H-group structure on p() with respect to which the inclusion p() i is an H-subgroup of an H-group (, µ) where map p be an H-transversal. Mathematics Subject Classification: 20J99 Keywords: H-transversal; H-subgroup; H-group 1 Introduction Let be a group with identity e. A map p : satisfying the following properties (i) p(e) = e (ii) p 2 = p (iii) p(g 1 g 2 ) = p( p(g 1 )g 2 ) is called a p-map. Let H be a subgroup of and S be a right transversal to H in. Then = HS. Thus each element g of can be uniquely written as

2 706 Swapnil Srivastava and Punish Kumar hx where h H and x S. Suppose x, y S and h H. Then a map p : defined by p(g) = x is a p-map. For a p-map on, the subset H = {g : p(g) = e} of is a subgroup of and the subset S = { p(g) : g } is a right transversal (with identity) of H in [5]. Ramji Lal [4] in his paper Transversals in roups have studied transversals in much more detail. Ungar and Foguel [3] has also given a way of decomposition of a group through an involution of a group into a twisted subgroup and a subgroup. In this paper, using homotopy theory and taking p to be continuous, we have defined the concept of H-subgroup and H-transversal. We have shown that there is a canonical H-group structure on p() with respect to which the inclusion p() i is an H-subgroup of an H-group (, µ) where map p be an H-transversal. Note: Throughout the paper represents homotopy between two maps. 2 H-Space In the present section, we have defined topological group, H-space, H-group, H-map, H-subgroup etc [1,2]. Definition 2.1. A topological group is a group that is also a topological space, satisfying the requirements that the map of into sending x y into x.y, and the map of into sending x into x 1, are continuous. Definition 2.2. A nonempty topological space with a base point is called a pointed topological space. Definition 2.3. A pointed topological space with base point e 0 together with a continuous multipication µ : for which the unique constant map c : defined by c(x) = e 0, is a homotopy identity, that is, each composite (c 1) µ and (1 c) µ is homotopic to identity map (1 : ), is called an H-space. Definition 2.4. Let be an H-space. The continuous multipication µ : is said to be homotopy associative if the following diagram (µ 1) (1 µ) µ µ is homotopy commutative i.e µ (µ 1) µ (1 µ).

3 H-transversals in H-groups 707 Definition 2.5. Let be an H-space. A continuous function φ : is called a homotopy inverse for and µ if each of the composites (φ 1) µ and (1 φ) µ is homotopic to homotopic identity c :. Definition 2.6. A homotopy associative H-space with a homotopy inverse satisfies the group axioms upto homotopy. Such a pointed space is called an H- group. Example 2.7. Any topological group is an H-group. Definition 2.8. The continuous multipication µ : in an H-group is said to be homotopy abelian if the diagram T µ µ is commutative upto homotopy i.e µ T µ where T (p 1, p 2 ) = (p 2, p 1 ). Definition 2.9. An H-group with homotopy abelian multiplication is called an abelian H-group. Definition If and are H-groups with multiplication µ and µ respectively. A continuous map α : is called a homomorphism if the diagram (α,α) µ α µ is commutative upto homotopy. Definition If and are H-groups with multiplication µ and µ respectively. A homomorphism α : is called an H-map if the diagram α c α c

4 708 Swapnil Srivastava and Punish Kumar is commutative upto homotopy that is α c c α where c and c homotopy identity for and respectively. are Definition An equivalence class of monomorphism in the category of H- groups is called an H-subgroup. More explicitly, let (, µ) be an H-group. An H-subgroup is an H-group (K, µ) together with an H-map φ : K if given any H-group (L, η) and two H-maps f 1, f 2 : L K such that φ f 1 φ f 2 f 1 f 2. Thus [φ] is a class of monomorphisms in the category of H-groups (objects are H-groups and morphisms are equivalence class of H-maps). This described a subgroup as an equivalence class of H-maps. Definition Consider the set S = {φ : K is an H-map : (K, ν) is an H-subgroup of an H-group (, µ)}.define two H-maps φ 1 : K 1 and φ 2 : K 2 equivalent if there exists H-maps h 1 : K 1 K 2 and h 2 : K 2 K 1 such that h 2 h 1 I K1, h 1 h 2 I K2, φ 2 h 1 φ 1 and φ 1 h 2 φ 2. Proposition Let (X, x 0 ) and (Y, y 0 ) be two pointed topological spaces. Then ΩX = {ω : ω : I X is a loop based at x 0 } is an H-group with continuous multiplication µ. Similarly ΩY = {ω : ω : I Y is a loop based at y 0 } is an H-group with continuous multiplication ν. Let f : (Y, y 0 ) (X, x 0 ) is a continuous map.then (ΩY, ν) is an H-subgroup together with an H-map (ΩY, ν) Ωf (ΩX, µ). Proof: Let (Z, +) be any H-group and h 1, h 2 : (Z, +) ΩY are two H-maps given by h 1 (n) = σ n,h 2 (n) = τ n. Let σ, τ ΩY and f σ, f τ are in same path component of ΩX then we have (Z, +) h 1 ΩY Ωf ΩX and (Z, +) h 2 ΩY Ωf ΩX are path homotopic, that is, Ωf h 1 p Ωf h 2 because there is a path χ : I ΩX such that χ(0) = f σ and χ(1) = f τ Now define χ : Z I ΩX by χ(n, t) = (χ(t)) n Then χ(n, 0) = (χ(0)) n = (f σ) n = f σ n = (Ωf h 1 )(n) and χ(n, 1) = (χ(1)) n = (f τ) n = f τ n = (Ωf h 2 )(n) If f σ and f τ lies in the same path component then σ and τ also lies in the same path component. Now Ωf h 1 p Ωf h 1 f σ n p f τ n σ n p τ n h 1 (n) p h 2 (n) h 1 p h 2 Thus ΩY is an H-subgroup with H-map Ωf : ΩY ΩX.

5 H-transversals in H-groups H-Transversal In this section, by defining H-transversal, we have shown that there is a canonical H-group structure on p() with respect to which the inclusion p() i is an H-subgroup of (, µ). (theorem 3.2) Definition 3.1. An H transversal in an H-group (, µ) is a continuous identity preserving map p : such that (i) p 2 p (ii) µ p p p µ ( p I ) Theorem 3.2. Let (, µ) be an H-group with base point identity element e of the group.let p be an H-transversal in an H-group (, µ). Then there is a canonical H-group structure on p() with respect to which the inclusion p() i is an H-subgroup of (, µ). Proof: From the definition of H-transversal, we have µ p p p µ ( p I ) Thus there is a homotopy H : I such that H((g 1, g 2 ), 0) = µ( p(g 1 ), p(g 2 )) H((g 1, g 2 ), 1) = p(µ( p(g 1 ), g 2 )) for all g 1, g 2 Define a product ν : p() p() p() by ν( p(g 1 ), p(g 2 )) = H(( p(g 1 ), p(g 2 )), 1) = p(µ( p( p(g 1 )), p(g 2 ))) = p(µ( p(g 1 ), p(g 2 ))) = ( p µ)( p(g 1 ), p(g 2 )) We show that ( p(), ν) is an H-group. Since p and µ are continuous so is ν. Now, (i) Since is an H-group so the constant map c : given by c (g) = e is a homotopy identity that is, µ (c 1 ) is homotopic to identity map 1 and similarly µ (1 c ) is also homotopic to identity map 1 Now for p(g) p() (µ (c 1 )) p(g) = µ (c ( p(g)), p(g)) = µ(e, p(g)) Let c p() : p() p() denote constant map on p() defined by c p() ( p(g)) = p(e) = e. Replacing above by p(). We have (ν (c p() 1 p() ))( p(g)) = (ν((c p() ( p(g)), p(g)))) = ν(e, p(g)) = ν( p(e), p(g)) = ( p µ)( p(e), p(g)) = p(µ( p(e), p(g))) = p((µ (c p() 1 p() )) p(g)) = p((µ (c 1 )) p(g)) p(1 ( p(g))) p( p(g)) p(g)

6 710 Swapnil Srivastava and Punish Kumar 1 p() ( p(g)) Thus ν (c p() 1 p() ) 1 p() Similarly, ν (1 p() c p() ) 1 p() Thus c p() is homotopic identity for ( p(), ν) (ii) Let φ : be homotopy inverse for (, µ). So µ (φ 1 ) and µ (1 φ) are homotopic to homotopy identity c for. Now (ν (1 p() φ p() ))( p(g)) = ν( p(g), φ p() ( p(g))) = ν( p(g), p(g 1 )) for some g 1 = ( p µ)( p(g), p(g 1 )) = p(µ(( p(g), p(g 1 )) = p(µ(( p(g), φ p() ( p(g))) = p(µ(1 p() φ p() )( p(g))) p(c p() ( p(g))) p(e) e c p() ( p(g)) Thus, ν (1 p() φ p() ) c p() Similarly we can show that ν (φ p() 1 p() ) c p() Hence φ p() is homotopy inverse for p(). (iii) Associativity Since (, µ) is associative.so the following diagram is commutative upto homotopy µ 1 µ 1 µ µ that is, µ (µ 1 ) µ (1 µ) Now,replacing by p(), we have to show that ν (1 p() ν) is homotopic to ν (ν 1 p() ), that is, the following diagram is commutative upto homotopy. p() p() 1 p() ν p() p() ν 1 p() p() p() ν ν p() Then (ν (1 p() ν))( p(g 1 ), p(g 2 ), p(g 3 )) = ν( p(g 1 ), ν( p(g 2 ), p(g 3 ))) = ν( p(g 1 ), ( p µ)( p(g 2 ), p(g 3 ))) ( p µ)( p( p(g 1 )), p(µ( p(g 2 ), p(g 3 )))) [Since p 2 = p ] p (µ p p)( p(g 1 ), µ( p(g 2 ), p(g 3 ))) p ( p µ p 1 )( p(g 1 ), µ( p(g 2 ), p(g 3 )))

7 H-transversals in H-groups 711 ( p p) µ( p( p(g 1 )), µ( p(g 2 ), p(g 3 ))) p µ( p(g 1 ), µ( p(g 2 ), p(g 3 ))) p (µ (1 µ))( p(g 1 ), ( p(g 2 ), p(g 3 ))) p (µ (µ 1 ))(( p(g 1 ), p(g 2 )), p(g 3 )) p µ(µ( p(g 1 ), p(g 2 )), p(g 3 )) ( p µ)(µ( p( p(g 1 )), p( p(g 2 ))), p(g 3 )) ( p µ)((µ p p)( p(g 1 ), p(g 2 ), p(g 3 ))) ( p µ)(( p µ ( p 1 ))( p(g 1 ), p(g 2 ), p(g 3 ))) ( p µ)(( p µ)( p( p(g 1 )), p(g 2 )), p(g 3 )) ( p µ)(( p µ)( p(g 1 ), p(g 2 )), p(g 3 )) ν(ν( p(g 1 ), p(g 2 )), p(g 3 )) (ν (ν 1 p() ))( p(g 1 ), p(g 2 ), p(g 3 )) Thus ν (1 p() ν) ν (ν 1 p() ) Now, we show that inclusion map i : p() is an H-map. First, we show that the diagram i i p() p() ν µ p() i is commutative upto homotopy which proves that i is an homomorphism. We have (µ (i i))( p(g 1 ), p(g 2 )) = µ( p(g 1 ), p(g 2 )) Now, (i ν)( p(g 1 ), p(g 2 )) = i(ν( p(g 1 ), p(g 2 ))) = i(( p µ)( p(g 1 ), p(g 2 ))) = i( p(µ( p(g 1 ), p(g 2 ))) = p(µ( p(g 1 ), p(g 2 ))) p(µ( p( p(g 1 )), p(g 2 ))) [ Since p 2 = p ] p(µ(( p 1 )( p(g 1 ), p(g 2 )))) ( p µ p 1 )( p(g 1 ), p(g 2 )) (µ p p)( p(g 1 ), p(g 2 )) µ( p( p(g 1 )), p( p(g 2 ))) µ( p(g 1 ), p(g 2 )) (µ i i)( p(g 1 ), p(g 2 )) Thus i ν µ i i Now, to show that the homomrphism i is an H-map, we prove that the diagram p() c p() i c p() i

8 712 Swapnil Srivastava and Punish Kumar is commutative upto homotopy where c p() and c denote constants map on p() and respectively. (c i)( p(g)) = c (i( p(g))) = c ( p(g)) = c p() ( p(g)) = i(c p() ( p(g))) So, c i i c p() Hence inclusion i : p() is an H-map. Now for any H-group (L, η), we have L f 1 f2 p() i where f 1, f 2 are H-maps, therefore i f 1 i f 2 f 1 f 2. Thus [i] is an H- subgroup in the category of H-groups(with objects as H-groups and morphisms as equivalence class of H-maps). References [1] E.H.Spanier,Algebraic Topology Mcraw Hill Book Company, [2] J.R.Munkres, Topology: A First Course Prentice Hall, [3] T. Foguel, A. A. Ungar. Involutory decomposition of groups into twisted subgroups and subgroups, J.roup Theory 3(1),(2000), [4] R. Lal. Transversals in groups, J.Algebra 181,(1996), [5] S. Srivastava, P. Kumar, Extension of roups using p-maps, Comm. in Algebra, (communicated) Received: September 3, 2014