Nuclear properties of loop extensions

The objectives of this paper is to give a systematic investigation of extension theory of loops. A loop extension is (left, right or middle) nuclear, if the kernel of the extension consists of elements associating (from left, right or middle) with all elements of the loop. It turns out that the natural non-associative generalizations of the Schreier's theory of group extensions can be characterized by different types of nuclear properties. Our loop constructions are illustrated by rich families of examples in important loop classes.


Introduction
A loop L is an extension of the loop N by the loop K if N is a normal subloop of L and K is isomorphic to the factor loop L/N . Extension theory deals with the classification of all possible extensions of N by K and studies their properties. The related problems in group theory are completely solved by Schreier theory of group extensions, cf. [9], [10], [4], Chapter XII, §48-49, pp. 121-131. But for loops A. A. Albert and R. H. Bruck proved in 1944 that construction of loop extensions of N by K has much more degrees of freedom, namely the multiplication function between different cosets = N of N can be prescribed arbitrarily. An interesting class of loop extensions of groups by loops is introduced in [5], where the multiplication of the extended loop is determined by an analogous formula as in the Schreier theory of groups. This paper contains characterizations and constructions of examples for interesting subclasses of such loops; these loops are called Schreier loops. In recent papers [6] and [7] a non-associative extension theory of Schreier type is investigated in a broader context, namely there are given characterizations of right nuclear automorphism-free extensions of groups by quasigroups with right identity element. As a consequence of these results, it turns out that the extension of a normal subgroup by the factor loop is isomorphic to an automorphism-free Schreier loop if and only if the normal subgroup is right and middle nuclear and there exists a left transversal to the normal subgroup (through the identity element of the loop) which commutes with this subgroup.
The aims of this paper are to find algebraic characterizations of Schreier loops and to examine the limits of the non-associative generalization of Schreier theory of extensions. In §2 we give the necessary definitions and formulate the basic constructions. §3 is devoted to the discussion of the interrelation between nuclear properties of normal subgroups in a loop and the corresponding Schreier extensions. Particularly, we show that for any Schreier extension this normal subgroup is the middle and right nuclear but in the general case it is not left nuclear. In §4 we introduce the notion of Schreier decomposition and show that a Schreier decomposition of a loop L is uniquely determined by a middle and right nuclear normal subgroup G, an isomorphism of a loop K to the factor loop L/G and by a left transversal to G through the identity element. §5 is devoted to the study of automorphisms of middle and right nuclear normal subgroups of a loop induced by middle inner mappings by loop elements. All of these maps are inner automorphisms if and ony if there exists a left transversal through the identity element to the subgroup commuting with this subgroup. In §6 we investigate different properties of Schreier decompositions of a loop. We give characterizations of loops having automorphism-free, respectively factor-free Schreier decompositions. §7 is devoted to the study of Schreier loops which are Schreier decompositions of the same loop with respect to the same normal subgroup.

Preliminaries
A quasigroup L is a set with a binary operation (x, y) → x · y such that for each x ∈ L the left and the right translations λ x : y → λ x y = xy : L → L, respectively ρ x : y → ρ x y = yx : L → L are bijective maps. We define the left and right division operations on L by (x, y) → x\y = λ −1 x y, respectively (x, y) → x/y = ρ −1 y x for all x, y ∈ L. A quasigroup L is a loop if it has a identity element e ∈ L. The right inner mappings of a loop L are the maps ρ −1 yx ρ x ρ y : L → L, x, y ∈ L. We will reduce the use of parentheses by the following convention: juxtaposition will denote multiplication, the operations \ and / are less binding than juxtaposition, and · is less binding than \ and /. For instance the expression xy/u · v\w is a short form of ((x · y)/u) · (v\w). The subgroups N l (L) = {u ∈ L; ux · y = u · xy, x, y ∈ L}, A loop L satisfies the left, respectively the right inverse property if there exists a bijection x → x −1 : L → L such that x −1 · xy = y, respectively yx · x −1 = y holds for all x, y ∈ L. The left alternative, respectively right alternative property of L is defined by the identity x · xy = x 2 y, respectively , respectively z(xy · x) = (zx · y)x for all x, y, z ∈ L. Any left (respectively right) Bol loop has the left (respectively right) alternative and inverse properties.  Let K and N be loops with identity elements ǫ ∈ K and e ∈ N , and let be a family of quasigroup multiplications on N such that the equations e▽ ǫ,α x = x and x▽ α,ǫ e = x are fulfilled for any α ∈ K and x ∈ N . The multiplication of the pairs (α, a), (β, b) ∈ K × N determines a loop L ▽ on K × N with identity (ǫ, e). Clearly, L ▽ is an extension of the normal subloopN = {(ǫ, a); a ∈ N } by the loop K, whereN is isomorphic to N .
In the following we will discuss nuclear properties of normal subgroups of loops and the corresponding extensions of groups by loops. The following lemma allows us to consider a family of extensions such that the normal subgroupḠ = {(ǫ, a); a ∈ G} is right nuclear but not middle or left nuclear.
Lemma 1 Let G be a group with identity e ∈ G, K a loop with identity ǫ ∈ K and let ψ σ : G → G be bijective maps depending on σ ∈ K satisfying ψ σ (ǫ) = e for any σ ∈ K and ψ ǫ = Id. The multiplication (i) The subgroupḠ is right nuclear.
(ii)Ḡ is middle nuclear if and only if ψ σ : G → G is an automorphism of G for any σ ∈ K.

(iii)Ḡ is left nuclear if and only if the map
Proof. The assertions (i) and (ii) can be obtained by direct computation.
Putting a = e the assertion follows.

Schreier extension
In the following we consider a special case of Bruck's extension process, assuming that the extended multiplication has an analogous expression as in Schreier theory of group extensions (cf. [5]). Let G be a group with identity e ∈ G, K a loop with identity ǫ ∈ K and let Aut(G) denote the automorphism group of G. If σ → Θ σ is a mapping K → Aut(G) with Θ ǫ = Id and f : together with the divisions The maps defined by (ǫ, t) → t :Ḡ → G and τ → (τ, e)Ḡ : K → L(Θ, f )/Ḡ are isomorphisms. Clearly, L(Θ, f ) is an extension of the groupḠ by the loop K.
L(Θ, f ) is a group if and only if K is a group and the identities are satisfied, (cf. [4], §48).

Lemma 3 For any Schreier loop
( i)Ḡ is middle and right nuclear, Proof. The assertions follow from Proposition 3.2, and Propositions 3.7, 3.8 and 3.10 in [5].

Schreier decomposition
Lemma 5 Let L be a loop extension of the group G by the loop K and let Proof. Since the map σ → (σ, e)G : K → L/G is an isomorphism the image of the coset (σ, e)G is the coset {F ((σ, e)(ǫ, s)) , s ∈ G} = {F(σ, e)s, s ∈ G} and the assertion follows.
We notice that the isomorphism F : L(Θ, f ) → L satisfies F(ǫ, t) = t for any t ∈ G if and only if F is an extension of the isomorphism I :Ḡ → G defined by I(ǫ, t) = t.
Definition 6 Let K and L be loops, G a normal subgroup of L and L(Θ, f ) a Schreier loop defined on K×G. A Schreier decomposition of L with respect to its normal subgroup G is an isomorphism F : The underlying isomorphism of the Schreier decomposition F is the map σ → F(σ, e)G : K → L/G.
The following lemma shows that by investigation of Schreier decompositions of L with respect to its normal subgroup G the middle and right nuclear property of G is a reasonable assumption. Left transversals to G in L Lemma 9 If G is a middle and right nuclear normal subgroup of the loop L then the maps T x | G : G → G, induced by the middle inner mappings Proof. If π : L → L/G is the canonical homomorphism then π(T x (t)) = π(x)π(t)/π(x) = ǫ, x ∈ L, t ∈ G, is called the Schreier loop corresponding to the data pair (κ, Σ).

Corollary 13 A loop L has a Schreier decomposition with respect to a normal subgroup G if and only if G is middle and right nuclear.
Proposition 14 If a loop L satisfies one of the following conditions: then any middle and right nuclear normal subgroup of L is nuclear.
Proof. It follows from Theorem 12 that for a middle and right nuclear normal subgroup G of L there is a Schreier decomposition F : L(Θ, f ) → L with respect to G. According to Propositions 3.7, 3.8, respectively 3.10 in [5] a Schreier loop L(Θ, f ) having the left inverse, left alternative, respectively flexible property, satisfies the condition (2). In this case we obtain from Proposition 3.2.(i) in [5] that the normal subgroupḠ = {(ǫ, t); t ∈ G} of L(Θ, f ) is nuclear, and hence G is also a nuclear subgroup of L.

Now, we
give examples for Schreier loops having the right Bol property such that the normal subgroupḠ is middle and right nuclear, but not nuclear. These properties can be verified by easy computation.
Example 15 Let K be a right Bol loop, G a group and H the group generated by the right inner mappings ρ −1 τ σ ρ σ ρ τ : K → K, σ, τ ∈ K. Let χ : H → G be a homomorphism such that the image χ(H) is not contained in the center of G. Define the maps f : K × K → G and Θ : K → Aut(G) by and consider the corresponding Schreier loop L(Id, f ).
Example 16 Let K and G be groups. Assume that the group K is not abelian and denote by K ′ the commutator subgroup of K. Let φ : K ′ → G be a homomorphism such that the image φ(K ′ ) is not contained in the center of G. Define the maps f : K × K → G and Θ : K → Aut(G) by and consider the corresponding Schreier loop L(Id, f ).
Example 17 Let K and G be groups with identity ǫ ∈ K and e ∈ G, respectively. Assume that the group K is not abelian. Let φ : K → G be a homomorphism such that the image φ(K) is not contained in the center of G. Define the maps f : K × K → G and Θ : K → Aut(G) by (σ), u ∈ G, σ ∈ K ι s (t) = sts −1 , s, t ∈ G and consider the corresponding Schreier loop L(ι φ , e).

Automorphisms of G induced by elements of L
According to Lemma 9 the maps T x | G are automorphisms of a middle and right nuclear normal subgroup G of a loop L, where x is an arbitrary element of L. If r ∈ G then T r | G is the inner automorphism ι r (t) = rtr −1 , r, t ∈ G.
Lemma 18 If G is a middle and right nuclear normal subgroup of a loop L then the automorphisms T xr | G and T rx | G with x ∈ L and r ∈ G can be decomposed as Proof. Since s and r belong to N r (L), we have T xr (s)·xr = xr ·s = x·ι r (s)r = xι r (s)·r = T x (ι r (s))x·r = T x (ι r (s))(xr), hence the first assertion is true. Similarly, the second assertion follows from since s ∈ N r (L) and T x (s), r ∈ N m (L). Proof. Assume that for any x ∈ L the map T x | G is an inner automorphism. Let Σ be a left transversal of L/G and g : Σ → G a map satisfying g(e) = e and T x | G = ι g(x) for any x ∈ Σ. Clearly, the set

Corollary 19 If G is middle and right nuclear normal subgroup in
is a left transversal of L/G. According to Lemma 18, and hence Σ * ⊂ C L (G). Conversely, let Σ be a left transversal of L/G such that T x | G = Id G for all x ∈ Σ. Any element of L is a product x · r with x ∈ Σ, r ∈ G and hence Lemma 18 yields that T x·r | G = T x | G • ι r = ι r , i.e. T x·r | G is an inner automorphism of G.
Lemma 21 For a middle and right nuclear normal subgroup G in L the mapping T| G : L → Aut(G) is a homomorphism if and only if G is nuclear.
Proof. For any s ∈ G, x, y ∈ L we have s ∈ N r (L), T y (s) ∈ N m (L) and hence T xy (s) · xy = x · T y (s)y = xT y (s) · y = T x (T y (s))x · y.
It follows that T| G : L → Aut(G) is a homomorphism if and only if for any x, y ∈ L, s ∈ G one has T x (T y (s))x · y = T x (T y (s)) · xy. Since T x | G , T y | G : G → G are bijective maps, the map T| G : L → Aut(G) is a homomorphism if and only G is left nuclear.
It follows from Proposition 14 the following Proof. Using the second formula of (4) we obtain that the Schreier loop defined by (4) is factor-free if and only if the map l : K → L satisfies l στ \l σ l τ = e for any σ, τ ∈ K, and hence the map l : K → L is a loop homomorphism. It follows that L has a factor-free Schreier decomposition if and only if there exists a left transversal Σ of L/G which is a subloop of L.
The following assertion shows the alteration of the Schreier decomposition of a loop L with respect to a normal subgroup G, if we change the underlying isomorphism.
Proposition 26 Let L(Θ, f ) be a Schreier decomposition of L with respect to G with underlying isomorphism κ : K → L/G and let µ be an automor- Proof. We denote the multiplication of L( Θ,f ) by• and define the map Theorem 27 The mapsΘ : K → Aut(G) andf : K × K → G determined by the left transversal Σ(l) = {l σ n(σ) ∈ κ(σ), σ ∈ K} can be expressed bȳ