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
Over the past two decades, a non-associative extension theory of binary systems has attracted considerable attention, especially from the viewpoints of construction and investigation of quasigroups and loops with special features (e.g. [2,3,8,[10][11][12]18,19,27,28]). Among these, a non-associative generalization of the group theoretical Schreier extension (cf. [29]) seems to be the simplest and serves as a prototype. A loop L is an extension of a normal subloop N by the loop K if K is isomorphic to the factor loop L/N , the normal subloop is called the kernel subloop of the extension. Equivalently, there is a short exact sequence of loops 1 → M → L → K → 1, (cf. [6], pp. [35][36][37][38][39][40][41][42][43]). Albert's description of general loop extensions yields a canonical form of the extended loop expressed with the help of a left transversal to the normal subgroup, any extension can be represented in Albert's form.
The objectives of this paper are to give a systematic investigation of nuclear properties of extensions, to find algebraic characterizations of their different types and to examine the limits of the non-associative generalization of theory of extensions. The first section contains the introduction and the necessary definitions and the basic constructions. Section 2 is devoted to the general theory of loop extensions and to the discussion of their nuclear properties. Section 2.1 examines the relationship between the Bruck's and Albert's description of extensions. Section 2.2 contains a discussion of the internal determination of an Albert extension by a triple of a normal subgroup, a loop isomorphism to the factor loop and a left transversal to the kernel subgroup. Section 2.3 is devoted to the classification of extensions of order 6 and to the description of their nuclear properties. Section 2.4 gives a characterization of right nuclear extensions in Albert's form, in this case the multiplication function between cosets of the kernel subgroup is a left multiplication in the kernel loop. Section 2.5 contains a construction of right nuclear but not nuclear extensions having left or right inverse property. In Sect. 3 we investigate right and middle nuclear extensions, called the Schreier extensions, which are the closest analogues of the group theoretical case. Section 3.1 is devoted to the construction of large classes of non-nuclear Schreier extensions satisfying the right Bol identity, using an extension method of Bol loops developed in [28] for classification of compact Bol loops. In Section 3.2 a discussion of internal constructions of Schreier extensions is given. In this study, the conjugation of the kernel subgroup by elements of the loop are an interesting tool. Section 3.3 is devoted to the investigation of automorphism-free and factor-free Schreier extensions by analogy in group theory and to the discussion of changing an extension by changing the underlying isomorphism or the left transversal of the decomposition. In Sect. 3.4 we investigate the relationship between different Schreier extensions of a loop.

Preliminaries
A quasigroup L is a set with a multiplication map (x, y) → x · y : L × L → L such that for each x ∈ L the left translations λ x : L → L, λ x y = xy, and the right translations ρ x : L → L, ρ x y = yx, are bijective maps. The left and right division operations on L are defined by the maps (x, y) → x\y = λ −1 x y, respectively (x, y) → x/y = ρ −1 y x, x, y ∈ L. An element e ∈ L is called left (right) identity if it satisfies e · x = x (x · e = x) for any x ∈ L. A left and right identity is called identity element. A quasigroup L is a loop if it has an identity element. The right inner mappings of a loop L are the maps ρ −1 yx ρ x ρ y : L → L, x, y ∈ L, the group generated by the right inner mappings is the right inner mapping group. The automorphism group of of a loop L is denoted by Aut(L).
We will reduce the use of parentheses by the following convention: juxtaposition will denote multiplication, the division operations are less binding than juxtaposition, and the multiplication symbol is less binding than the divisions. For instance the expression xy/u·v\w is a short form of ((x·y)/u)·(v\w). The subgroups 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. A loop with left and right inverse property has inverse property. The left alternative, respectively right alternative property of L is defined by the identity Lemma 2 (Bruck [4], Theorem 10B, p. 168 and [5], p. 779.). Assume that a loop L with identity element e ∈ L is an extension of a normal proper subloop N by a loop K. Then L is isomorphic to the loop K × N defined on K × N by the multiplication where (a, b) → α,β (a, b), α, β ∈ K are quasigroup multiplications on N satisfying the conditions (i) , coincides with the multiplication of N , (ii) e ∈ N is a left identity element of the multiplications ,α , (iii) e ∈ N is a right identity element of the multiplications α, , α ∈ K. Conversely, if for any α, β ∈ K there is given a quasigroup multiplication α,β on a loop N satisfying the conditions (i), (ii), (iii), then the multiplication (1) determines a loop K × N which is an extension of the subloop { } × N by the loop K.
(2) determines a Bruck extension K × N of the loop N by the loop K and the map Φ : Proof. If π : L → L/N is the canonical homomorphism then we have π (l ξη \(l ξ x · l η y)) = N and hence ξ,η (x, y) is contained in N . The conditions i, ii, iii of Lemma 2 are routinely verified. One has Φ((ξ, x)(η, y)) = l ξ x · l η y = Φ(ξ, x)Φ(η, y) and hence the assertion follows.
Remark 5. Let Σ be a left transversal to N in L and K × N the Bruck extension determined by the quasigroup multiplications (2). Then (iii ) e ∈ N is the identity element of the multiplications α, , α ∈ K.
It follows from Lemma 4 and Remark 5, that any Bruck extension K × N is isomorphic to an Albert extension. Now, we construct such an isomorphism.
hence e ∈ N is a left identity element of ,α . Moreover e ∈ N is the identity element of α, for any α ∈ K, since the identities * and from the right identity property of e ∈ N with respect to the multiplication α, . Hence K × * N is an Albert extension.
with respect to Σ coincide with the multiplications α,β for any α, β ∈ K.

Albert Decompositions
Let L be a loop, N a normal subloop of L and let K × N be an Albert extension of N by K.
Proof. The assertion follows from the observation that a loop isomorphism For a data triple (N, κ, Σ) of an Albert decomposition of L we define the maps Proof. An element x ∈ L can be uniquely decomposed as a product x = l σ s with l σ ∈ Σ, s ∈ N . The bijective map F : Consequently F : L(N, κ, Σ) → L is an Albert decomposition of L and κ is the underlying isomorphism. Proof. We replace in K ×N the elements ( , t) ∈ { }×N by the corresponding elements t ∈ N and define a loop L on the set

Albert Extensions of Order 6
Extensions of C 2 by C 3 . Since the loops of order 2 or 3 are cyclic groups, the Albert extensions of order 6 are extensions of a cyclic group by a cyclic group. We investigate the Albert extension loop L of a normal subgroup C 2 = {0, 1} by the group C 3 = {0, 1, 2} using the construction described in Definition 4. A quasigroup of order two is either C 2 or it is given by the multiplication According to Theorem 2.1. in [26] {0} × C 3 is contained in the center of C 3 , i.e. all elements of {0} × C 3 commute and associate with all other elements of L f , hence the extension is nuclear. Since γ) for any α, β, γ ∈ C 3 . Investigating all of the cases α, β, γ ∈ {1, 2}, we find that L f is a group if and only if the mapping ϕ is an isomorphism L f → L f and we have the isomorphisms Clearly, L f0 , L f1 , L f6 , L f7 , L f9 and L f15 are commutative.
Hence we get i.e. the function f * satisfies f * (1, 1) = f (1, 1) and f * (α, β) = f (α, β) + 1 if α, β ∈ {1, 2} and α = 2 or β = 2. It follows that we have the isomorphisms: from the further investigation. L f0 is isomorphic to the cyclic group C 6 . L f1 and L f2 are commutative and non-commutative proper loops. We obtain the following Extensions of C 3 by C 2 . In the following we identify C 3 with the additive group of the field F 3 of order 3. Consider the multiplication table (Latin square) of a quasigroup. Its rows (columns) are permutations describing the left (right) translations. The even permutations of F 3 are the maps x → x + q and the odd permutations are x → 2x+q with some q ∈ F 3 , since x → 2x is a transposition. Consequently any permutation of F 3 can be expressed as x → (1+σ)x+q with σ ∈ {0, 1} ⊂ F 3 and q ∈ F 3 , this permutation is even for σ = 0 and odd for σ = 1. It is easy to check that the rows (columns) of the multiplication table are permutations of the same parity. For a given parity of rows (columns), the tables can only differ in the order of the rows (columns). Hence if x → (1 + σ)x + q is the permutation given by the first row (column), then the i-th row (column), i = 2, 3, is the permutation It follows that a quasigroup multiplication defined on F 3 can be expressed by Let (α, a)(β, b) = (α+β, α,β (a, b)), α, β ∈ C 2 , a, b ∈ F 3 , be the multiplication of an Albert extension of C 3 by C 2 . Then for any a, b ∈ F 3 . A loop of order 3 is necessarily a group, hence 1,0 (a, b) = a+b. The quasigroup with multiplication 0,1 (a, b) is isomorphic either to C 3 or 0,1 (a, b) = 2a + b, hence 0,1 (a, b) = (1 + θ)a + b with θ ∈ {0, 1} ⊂ F 3 . According to (6) we can express For ξ ∈ C 2 = {0, 1} denote ξ ∈ F 3 the canonical copy of elements of {0, 1} in F 3 , then (ξ + η) = ξ + η + ξ η for any ξ, η ∈ C 2 = {0, 1}.

Right Nuclear Extensions
In the following we discuss nuclear properties of normal subloops of loops. Since the left, right or middle nuclei of a loop are necessarily groups, we will investigate extensions of groups G by loops K. In the following Sym(M ) denotes the group of all permutations of a set M.
Definition 18. Let Γ : K × K → Sym(G) be a map satisfying Γ ,σ (e) = e and Γ σ, = Id for all σ ∈ K. We denote by K × Γ G the right nuclear Albert and let K × Γ G be the Albert extension determined by the multiplications Proof. Applying the natural homomorphism L → L/G to l ξη \(l ξ x · l η ) we get that Γ ξ,η (x) ∈ G for any ξ, η ∈ K, x ∈ G, hence Γ ξ,η ∈ Sym(G). The bijective map Φ : Hence we obtain the assertion.

Right Nuclear Extensions with Left or Right Inverse Property
Let K × Γ G be a right nuclear Albert extension of the group G by the group K. In the following we investigate the fulfilment of the left, respectively right inverse property in K × Γ G with the additional condition that the left, respectively right inverses are of the form (α −1 , a −1 ).
Lemma 24. A right nuclear Albert extension K × Γ G has the left inverse property with left inverses (α −1 , a −1 ), α ∈ K, a ∈ G, if and only if Γ is equivariant under the action of π.
Example 25. Define the map Γ : K × K → Sym(G) by Γ α −1 ,α = Γ α, = Id on Ξ. Choose the value Γ σ,τ ∈ Sym(G) for an element of any orbit of the group Π in K × K \ Ξ arbitrarily and determine the map Γ assuming that Γ is equivariant under Π. The obtained right nuclear Albert extensions K × Γ G are neither middle nor left nuclear, if some values of Γ are not belonging to the set These examples show that there are many right nuclear Albert extensions having the left inverse property which are neither middle nor left nuclear. In contrast to the left inverse property the right inverse property is rather restrictive for right nuclear Albert extensions.

Corollary 27.
A right nuclear Albert extension K × Γ G having the right inverse property with right inverses (α −1 , a −1 ), α ∈ K, a ∈ G, is middle nuclear, too.

Schreier Extensions
In the following we will investigate right and middle nuclear Albert extensions determined by a map Γ : K × K → Sym(G). According to Theorem 21 (i) in this case the map Γ has the form Γ σ,τ (s) = f (σ, τ )Θ τ (s), where Θ τ ∈ Aut(G) for any τ ∈ K. Definition 28. A right and middle nuclear Albert extension is called Schreier extension. A Schreier extension determined by the map Γ σ,τ (s) = f (σ, τ )Θ τ (s) will be denoted by K × Θ f G.