loops/doc/chap2.txt
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2 Mathematical Background
We assume that you are familiar with the theory of quasigroups and loops,
for instance with the textbook of Bruck [Bru58] or Pflugfelder [Pfl90].
Nevertheless, we did include definitions and results in this manual in order
to unify terminology and improve legibility of the text. Some general
concepts of quasigroups and loops can be found in this chapter. More special
concepts are defined throughout the text as needed.
2.1 Quasigroups and Loops
A set with one binary operation (denoted ⋅ here) is called groupoid or
magma, the latter name being used in GAP.
An element 1 of a groupoid G is a neutral element or an identity element if
1⋅ x = x⋅ 1 = x for every x in G.
Let G be a groupoid with neutral element 1. Then an element x^-1 is called a
two-sided inverse of x in G if x⋅ x^-1 = x^-1⋅ x = 1.
Recall that groups are associative groupoids with an identity element and
two-sided inverses. Groups can be reached in another way from groupoids,
namely via quasigroups and loops.
A quasigroup Q is a groupoid such that the equation x⋅ y=z has a unique
solution in Q whenever two of the three elements x, y, z of Q are specified.
Note that multiplication tables of finite quasigroups are precisely latin
squares, i.e., square arrays with symbols arranged so that each symbol
occurs in each row and in each column exactly once. A loop L is a quasigroup
with a neutral element.
Groups are clearly loops. Conversely, it is not hard to show that
associative quasigroups are groups.
2.2 Translations
Given an element x of a quasigroup Q, we can associative two permutations of
Q with it: the left translation L_x:Q-> Q defined by y↦ x⋅ y, and the right
translation R_x:Q-> Q defined by y↦ y⋅ x.
The binary operation xbackslash y = L_x^-1(y) is called the left division,
and x/y = R_y^-1(x) is called the right division.
Although it is possible to compose two left (right) translations of a
quasigroup, the resulting permutation is not necessarily a left (right)
translation. The set {L_x|x∈ Q} is called the left section of Q, and {R_x|x∈
Q} is the right section of Q.
Let S_Q be the symmetric group on Q. Then the subgroup Mlt_λ(Q)=⟨ L_x|x∈ Q⟩
of S_Q generated by all left translations is the left multiplication group
of Q. Similarly, Mlt_ρ(Q)= ⟨ R_x|x∈ Q⟩ is the right multiplication group of
Q. The smallest group containing both Mlt_λ(Q) and Mlt_ρ(Q) is called the
multiplication group of Q and is denoted by Mlt(Q).
For a loop Q, the left inner mapping group Inn_λ(Q) is the stabilizer of 1
in Mlt_λ(Q). The right inner mapping group Inn_ρ(Q) is defined dually. The
inner mapping group Inn(Q) is the stabilizer of 1 in Q.
2.3 Subquasigroups and Subloops
A nonempty subset S of a quasigroup Q is a subquasigroup if it is closed
under multiplication and the left and right divisions. In the finite case,
it suffices for S to be closed under multiplication. Subloops are defined
analogously when Q is a loop.
The left nucleus Nuc_λ(Q) of Q consists of all elements x of Q such that
x(yz) = (xy)z for every y, z in Q. The middle nucleus Nuc_μ(Q) and the right
nucleus Nuc_ρ(Q) are defined analogously. The nucleus Nuc(Q) is the
intersection of the left, middle and right nuclei.
The commutant C(Q) of Q consists of all elements x of Q that commute with
all elements of Q. The center Z(Q) of Q is the intersection of Nuc(Q) with
C(Q).
A subloop S of Q is normal in Q if f(S)=S for every inner mapping f of Q.
2.4 Nilpotence and Solvability
For a loop Q define Z_0(Q) = 1 and let Z_i+1(Q) be the preimage of the
center of Q/Z_i(Q) in Q. A loop Q is nilpotent of class n if n is the least
nonnegative integer such that Z_n(Q)=Q. In such case Z_0(Q)le Z_1(Q)le dots
le Z_n(Q) is the upper central series.
The derived subloop Q' of Q is the least normal subloop of Q such that Q/Q'
is a commutative group. Define Q^(0)=Q and let Q^(i+1) be the derived
subloop of Q^(i). Then Q is solvable of class n if n is the least
nonnegative integer such that Q^(n) = 1. In such a case Q^(0)ge Q^(1)ge ⋯ ge
Q^(n) is the derived series of Q.
2.5 Associators and Commutators
Let Q be a quasigroup and let x, y, z be elements of Q. Then the commutator
of x, y is the unique element [x,y] of Q such that xy = [x,y](yx), and the
associator of x, y, z is the unique element [x,y,z] of Q such that (xy)z =
[x,y,z](x(yz)).
The associator subloop A(Q) of Q is the least normal subloop of Q such that
Q/A(Q) is a group.
It is not hard to see that A(Q) is the least normal subloop of Q containing
all commutators, and Q' is the least normal subloop of Q containing all
commutators and associators.
2.6 Homomorphism and Homotopisms
Let K, H be two quasigroups. Then a map f:K-> H is a homomorphism if f(x)⋅
f(y)=f(x⋅ y) for every x, y∈ K. If f is also a bijection, we speak of an
isomorphism, and the two quasigroups are called isomorphic.
An ordered triple (α,β,γ) of maps α, β, γ:K-> H is a homotopism if α(x)⋅β(y)
= γ(x⋅ y) for every x, y in K. If the three maps are bijections, then
(α,β,γ) is an isotopism, and the two quasigroups are isotopic.
Isotopic groups are necessarily isomorphic, but this is certainly not true
for nonassociative quasigroups or loops. In fact, every quasigroup is
isotopic to a loop.
Let (K,⋅), (K,∘) be two quasigroups defined on the same set K. Then an
isotopism (α,β, id_K) is called a principal isotopism. An important class of
principal isotopisms is obtained as follows: Let (K,⋅) be a quasigroup, and
let f, g be elements of K. Define a new operation ∘ on K by x∘ y =
R_g^-1(x)⋅ L_f^-1(y), where R_g, L_f are translations. Then (K,∘) is a
quasigroup isotopic to (K,⋅), in fact a loop with neutral element f⋅ g. We
call (K,∘) a principal loop isotope of (K,⋅).