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Notation
Notation
What follows is a list of symbols and other notation used in this book. Page references guide the reader to the place where the synnbol is defined or first mentioned.
Symbol
Meaning
P
the prime numbers
N
the natural numbers, 1,2,3,...
Z
the integers
Q
the rational numbers
R
the real numbers
C
the complex numbers
Fp
the field of p elements, p a prime
(j)
the Euler function, p. 175
^„
unless specifically stated to the contrary, this always denotes a primitive nth root of unity, p. 175, p. 248
(F, Qf,/^)
a generalized quaternion algebra over a field F, p. 17
»(F)
( F , - l , - l ) , p . 17
H(R)
(R,—1,—1), the classical division algebra of real quaternions, p. 17
(F, Of,/?, 7)
a Cayley-Dickson algebra over a field F , p. 17
C
the Cayley numbers, p. 15
Mn{R)
the ring of n x n matrices over a ring R
383
384
NOTATION
R[Xi,
X 2 , . . . , Xn]
the ring of polynomials in c o m m u t i n g indeterminates X\^. . . , Xn with coefficients in a ring R
R[Xi, X f \ . . . , Xn, X~^]
the ring of Laurent polynomials in c o m m u t i n g indeterminates X i , . . . , X n , p. 153
R^
the set of ordered n-tuples over a ring R
RL
the loop ring of a loop L over a ring R, p. 85
M 04) A^
the tensor product of modules M and A'' over a c o m m u t a t i v e ring with unity, p. 39
Cn
the cyclic group of order n
Dn
the dihedral group of order 2n
Qg
the quaternion group of order 8
Mi6{Qs)
the Cayley loop, p. 68
Z{p^)
the p-Priifer group, p. 358
M(G,*,^o)
the Moufang loop determined by a nonabelian group G with involution * and central element (70, p. 80
M{G, - 1 ,
.9(1
the special case of M(G,^,
go) where * is the in-
verse m a p on G, p. 81
M(G, 2)
M ( r ; , - l , l ) , p. 80
In
where / is an ideal of a ring R, and n G N, this is {r e R\ rire
Lp
/ } , p. 157
where L is a loop (or group) and p is a prime, this is the set of elements of L whose order is finite and a power of p, p. 157
^p'
where L is a loop (or group) and p is a {)rime, this is the set of elements of L of finite order relatively prime to p, p. 157
[E: F]
where E/F
is a field extension, this is the dimen-
sion of E" as a vector space over F , p. 175 G?i\{E/F)
the Galois group of a Galois field extension
E/F,
p. 248 F{a)
where F is a field and a is an element contained in some extension field of K of F , this is the smallest subfield of K containing F and cv, p. 45
NOTATION
Z[a]
385
the subring of Q(of) generated by an elennent a E C, p. 202
J\fil{R)
the upper nil radical of a ring R, p. 148
V{R)
the prime radical of a ring R, p. 148
J[R)
the Jacobson radical of a ring R, p. 148
GL(2, R)
the general linear group of degree 2 over a ring i^, p. 82
SL(2, R)
the special linear group of degree two over a ring R, p. 82
PSL(2, R)
the projective special linear group of degree two over a ring R^ p. 215
3(/^)
Zorn's vector m a t r i x algebra over a ring R, p. 21
GLL(2, R)
the 2 X 2 invertible elements of 3(7?.), p. 83
SLL(2, R)
the elements of GLL(2, R) of determinant 1, p. 83
Sdet=±\
the matrices of determinant 1 in a subloop S of GLL(2,Z[
'^det=±i
^^^ quotient loop 5'det=±i/{/, — ^}) where / is the identity m a t r i x of Zorn's vector matrix algebra, p. 317
5'^i(det)=±i
those matrices in a subloop S of GLL(2, Z[<^n]) with the property t h a t the norm of dei A is ibl, p. 316
^nidet^-±\
^^^ quotient loop 5'n(det)=±i/{^, — 0 , whcre / is the identity matrix of Zorn's vector m a t r i x algebra, p. 316
x^
the right inverse of an element x in a loop, p. 50
x^
the left inverse of an element x in a loop, p. 50
R[x)
right translation by x, p. 50
L[x)
left translation by x, p. 50
T{x)
R{x)L{x)-\p. 57
R{x,y)
R{x)R{y)R{xy)-\p.b7
L[x,y)
L{x)L{y)L{yx)-\p.b7
(a, 6)
the c o m m u t a t o r of elements x and y in a loop, p. 52
386
{X,Y)
NOTATION
for subsets X and F of a loop, this is the set of all c o m m u t a t o r s (x, y), x ^ X^ y G V, p . 52
{a,b,c)
the associator of elements a, 6 and c in a loop, p. 52
(X,Y,Z)
for subsets X , y , Z of a loop, this is the set of all associators (ic, y^z)^ x ^ X ^ y ^Y
^ z E. Z ^ p. 52
[a,b]
the c o m m u t a t o r of elements a and 6 in a ring, p. 6
[X,Y]
for subsets X and F of a ring, this is the set of all c o m m u t a t o r s [x, y], x E -^, y G F , p. 6
[a, b, c]
the associator of elements a, 6, c in a ring, p. 6
[X, Y, Z]
for subsets X , F , Z of a ring, this is the set of all associators [x, y, z], x G ^ , y G F , z £ Z^ p. 6
L'
the commutator-associator subloop of a loop L, p. 53
Z{X)
the centre of a loop or ring X , p. 6 and 53
M{X)
the nucleus of a loop or ring X , p. 6 and 53
M{L)
the multiplication group of a loop L, p. 50
Inn(L)
the inner mapping group of a loop L, p. 57
[L: / / ]
where // is a subloop of an inverse property loop L, this is the index of H in L, p. 62
Aut(L)
the group of automorphisms of a loop L
CM
where A^ is a normal subloop of a loop L, this is the kernel of the m a p RL —> R[L/N],
which is
the linear extension of the natural m a p L —^ L/N, p. 149 6
ei^ where L is a loop, this denotes the augmentation m a p RL -^ R, p. 149
A ( L , N)
where L is a loop with normal subloop TV, this is kere^v, p. 149
A(L)
for a loop L, this is A ( L , L ) , the augmentation ideal of a loop ring RL, p. 149
N
where N is a finite loop contained in a loop ring, ^his is E n G N ^ ' P- 1^1
NOTATION
387
N
where N is a. finite loop contained in a loop ring RL with \N\ invertible in i?, this is AnN^ p. 151
^
{9) — Yl7=o 5'*' where g is an element of finite order n in a loop ring, p. 200
p
(^) rr 1 Yl^=o 9^' where g is an element of finite order n in a loop ring RL^ with n invertible in R^ p. 218
U{R)
the set of units in a ring 7?, p. 81
U{RL)
the set of units in the loop ring RL
U\{RL)
the set of normalized units in the loop ring RL^ p. 236
TU{RL)
the set of torsion units in the loop ring RL, p. 236
TU\ [RL)
the set of normalized torsion units in the loop ring RL, p. 236
char R,
the characteristic of a ring R, p.88
p[A)
the rank of an abelian group A] that is, the minimal number of generators for A, p. 132
o(x)
the order of an element x in a loop, p. 128
ind2(A)
an invariant of a field K related to its 2-sequence, p. 265 this is 1 if x^ 4- 1 == 0 has a solution in the field K and 0 otherwise, p. 265
0[K) i{K)
this is 1 if x^ -fy^ = —1 has a solution in the field K and 0 otherwise, p. 265
c[K)
this is 1 if x^ -h y^ -\- P -f tf;^ = _ i has a solution in the field K and 0 otherwise, p. 282
What follows is a list of symbols and other notation used in this book. Page references guide the reader to the place where the synnbol is defined or first mentioned.
Symbol
Meaning
P
the prime numbers
N
the natural numbers, 1,2,3,...
Z
the integers
Q
the rational numbers
R
the real numbers
C
the complex numbers
Fp
the field of p elements, p a prime
(j)
the Euler function, p. 175
^„
unless specifically stated to the contrary, this always denotes a primitive nth root of unity, p. 175, p. 248
(F, Qf,/^)
a generalized quaternion algebra over a field F, p. 17
»(F)
( F , - l , - l ) , p . 17
H(R)
(R,—1,—1), the classical division algebra of real quaternions, p. 17
(F, Of,/?, 7)
a Cayley-Dickson algebra over a field F , p. 17
C
the Cayley numbers, p. 15
Mn{R)
the ring of n x n matrices over a ring R
383
384
NOTATION
R[Xi,
X 2 , . . . , Xn]
the ring of polynomials in c o m m u t i n g indeterminates X\^. . . , Xn with coefficients in a ring R
R[Xi, X f \ . . . , Xn, X~^]
the ring of Laurent polynomials in c o m m u t i n g indeterminates X i , . . . , X n , p. 153
R^
the set of ordered n-tuples over a ring R
RL
the loop ring of a loop L over a ring R, p. 85
M 04) A^
the tensor product of modules M and A'' over a c o m m u t a t i v e ring
Cn
the cyclic group of order n
Dn
the dihedral group of order 2n
Qg
the quaternion group of order 8
Mi6{Qs)
the Cayley loop, p. 68
Z{p^)
the p-Priifer group, p. 358
M(G,*,^o)
the Moufang loop determined by a nonabelian group G with involution * and central element (70, p. 80
M{G, - 1 ,
.9(1
the special case of M(G,^,
go) where * is the in-
verse m a p on G, p. 81
M(G, 2)
M ( r ; , - l , l ) , p. 80
In
where / is an ideal of a ring R, and n G N, this is {r e R\ rire
Lp
/ } , p. 157
where L is a loop (or group) and p is a prime, this is the set of elements of L whose order is finite and a power of p, p. 157
^p'
where L is a loop (or group) and p is a {)rime, this is the set of elements of L of finite order relatively prime to p, p. 157
[E: F]
where E/F
is a field extension, this is the dimen-
sion of E" as a vector space over F , p. 175 G?i\{E/F)
the Galois group of a Galois field extension
E/F,
p. 248 F{a)
where F is a field and a is an element contained in some extension field of K of F , this is the smallest subfield of K containing F and cv, p. 45
NOTATION
Z[a]
385
the subring of Q(of) generated by an elennent a E C, p. 202
J\fil{R)
the upper nil radical of a ring R, p. 148
V{R)
the prime radical of a ring R, p. 148
J[R)
the Jacobson radical of a ring R, p. 148
GL(2, R)
the general linear group of degree 2 over a ring i^, p. 82
SL(2, R)
the special linear group of degree two over a ring R, p. 82
PSL(2, R)
the projective special linear group of degree two over a ring R^ p. 215
3(/^)
Zorn's vector m a t r i x algebra over a ring R, p. 21
GLL(2, R)
the 2 X 2 invertible elements of 3(7?.), p. 83
SLL(2, R)
the elements of GLL(2, R) of determinant 1, p. 83
Sdet=±\
the matrices of determinant 1 in a subloop S of GLL(2,Z[
'^det=±i
^^^ quotient loop 5'det=±i/{/, — ^}) where / is the identity m a t r i x of Zorn's vector matrix algebra, p. 317
5'^i(det)=±i
those matrices in a subloop S of GLL(2, Z[<^n]) with the property t h a t the norm of dei A is ibl, p. 316
^nidet^-±\
^^^ quotient loop 5'n(det)=±i/{^, — 0 , whcre / is the identity matrix of Zorn's vector m a t r i x algebra, p. 316
x^
the right inverse of an element x in a loop, p. 50
x^
the left inverse of an element x in a loop, p. 50
R[x)
right translation by x, p. 50
L[x)
left translation by x, p. 50
T{x)
R{x)L{x)-\p. 57
R{x,y)
R{x)R{y)R{xy)-\p.b7
L[x,y)
L{x)L{y)L{yx)-\p.b7
(a, 6)
the c o m m u t a t o r of elements x and y in a loop, p. 52
386
{X,Y)
NOTATION
for subsets X and F of a loop, this is the set of all c o m m u t a t o r s (x, y), x ^ X^ y G V, p . 52
{a,b,c)
the associator of elements a, 6 and c in a loop, p. 52
(X,Y,Z)
for subsets X , y , Z of a loop, this is the set of all associators (ic, y^z)^ x ^ X ^ y ^Y
^ z E. Z ^ p. 52
[a,b]
the c o m m u t a t o r of elements a and 6 in a ring, p. 6
[X,Y]
for subsets X and F of a ring, this is the set of all c o m m u t a t o r s [x, y], x E -^, y G F , p. 6
[a, b, c]
the associator of elements a, 6, c in a ring, p. 6
[X, Y, Z]
for subsets X , F , Z of a ring, this is the set of all associators [x, y, z], x G ^ , y G F , z £ Z^ p. 6
L'
the commutator-associator subloop of a loop L, p. 53
Z{X)
the centre of a loop or ring X , p. 6 and 53
M{X)
the nucleus of a loop or ring X , p. 6 and 53
M{L)
the multiplication group of a loop L, p. 50
Inn(L)
the inner mapping group of a loop L, p. 57
[L: / / ]
where // is a subloop of an inverse property loop L, this is the index of H in L, p. 62
Aut(L)
the group of automorphisms of a loop L
CM
where A^ is a normal subloop of a loop L, this is the kernel of the m a p RL —> R[L/N],
which is
the linear extension of the natural m a p L —^ L/N, p. 149 6
ei^ where L is a loop, this denotes the augmentation m a p RL -^ R, p. 149
A ( L , N)
where L is a loop with normal subloop TV, this is kere^v, p. 149
A(L)
for a loop L, this is A ( L , L ) , the augmentation ideal of a loop ring RL, p. 149
N
where N is a finite loop contained in a loop ring, ^his is E n G N ^ ' P- 1^1
NOTATION
387
N
where N is a. finite loop contained in a loop ring RL with \N\ invertible in i?, this is AnN^ p. 151
^
{9) — Yl7=o 5'*' where g is an element of finite order n in a loop ring, p. 200
p
(^) rr 1 Yl^=o 9^' where g is an element of finite order n in a loop ring RL^ with n invertible in R^ p. 218
U{R)
the set of units in a ring 7?, p. 81
U{RL)
the set of units in the loop ring RL
U\{RL)
the set of normalized units in the loop ring RL^ p. 236
TU{RL)
the set of torsion units in the loop ring RL, p. 236
TU\ [RL)
the set of normalized torsion units in the loop ring RL, p. 236
char R,
the characteristic of a ring R, p.88
p[A)
the rank of an abelian group A] that is, the minimal number of generators for A, p. 132
o(x)
the order of an element x in a loop, p. 128
ind2(A)
an invariant of a field K related to its 2-sequence, p. 265 this is 1 if x^ 4- 1 == 0 has a solution in the field K and 0 otherwise, p. 265
0[K) i{K)
this is 1 if x^ -fy^ = —1 has a solution in the field K and 0 otherwise, p. 265
c[K)
this is 1 if x^ -h y^ -\- P -f tf;^ = _ i has a solution in the field K and 0 otherwise, p. 282