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Notation
Notation Number Systems
the natural numbers the rational integers the p-adic integers the rational numbers the real numbers the complex numbers the integers mod m
Set Theory
C
c 1x1
X-Y E
U~EIA~ ni~~Ai
Imf
proper inclusion inclusion the cardinality of the set X the complement of Y in X membership sign the union of the sets the intersection of the sets the image of the map f : X -t Y
Number Theory
a divides b a does not divide b the greatest common divisor of a and b the p p a r t of n the p'-part of n
Group Theory
N a G CG(X) NG(X) o(g) L7.
vl
the subgroup generated by X the cyclic group of order n direct product of G1 and Gz the subgroup of A generated by elements of order dividing p the subgroup of A generated by elements up, a E A N is a normal subgroup of G the centralizer of X in G the normalizer of X in G the order of g E G - 2-1 y -1 xy (unless explicitly stated otherwise) = [G, GI the commutator subgroup of G the center of G the group of all nonsingular linear transformations of V the subgroup of GL(V) consisting of elements of determinant 1 = GL(V)/F' - lv the group of all n x n invertible matrices over R = { A E GLn(R)ldet(A) = 1) = GLn (R)/Z(GLn (R)) = SLn(R)/Z(SLn(R))
Notation
G1@G2
x G(x) Aut(X) OP(G) OPl(G) 0, (GI o n 1 ...?Ti(G)/OT1...T,-l (G) AP g
A[nl
GYP) An sn
t(A) F* U(R)
AH @i€~Gi n i €Gi ~
(G : H) Hom(G, A) P(G1, G2, A ) P(G, A) GAG Kerp
= GLn (Fq ) = SLm(Fq) = PGLn(Fq) = PSLn(Fq) the uni triangular linear group the diagonal subgroup of GL,(q) the orthogonal group of V the special orthogonal group of V the pin group of V the spin group of V the symplectic group the n-th term of the lower central series of G the tensor product of groups GI and G2 the image of x under the action of g E G the stabilizer of x the automorphism group of X the maximal normal psubgroup of G the maximal normal pl-subgroup of G the maximal normal T-subgroup of G = OiTi (G/OTI ...Ti-] (G)) the p-component of A = {a E Alan = 1) p-commutator subgroup of G the alternating group of degrees n the symmetric group of degree n the torsion subgroup of A the multiplicative group of a field F the unit group of R = {a E Alha = a for all h E H ) direct sum of groups G; direct product of groups G; the index of H in G the group of all homomorphisms of G into A the group of all pairings of GI and G2 into A the group of all pairings of G into A = ( G @ G ) / < g @ g l gE G
the kernel of p
>
the a-covering group of G the socle of G the automorphism group of G the Frattini subgroup of G the free group freely generated by X the n-th term of the lower central series of G the dihedral group of order 2m, m 2 the generalized quaternion group of order 4m, m 2 the semidihedral group of order 2m, m >3 =< a , blapm-I = 1, bP = 1, bab-1 = a l + ~ " - 2> the quadratic group the Thompson subgroup of P the subgroup of anti-symmetric pairings the exponent of G the n-th dimension subgroup of G the minimal number of generators of G =< [x,y]lx € y € Y > = [ [ X I , . . ,xn-~],xn] = [ [ X I- ,.. 7 Xn-l],Xn] the free product of A and B the group of universal commutator relations the deficiency of G the second nilpotent product of A and B the wreath product of A and B =A = [A,B , . ..,B] ( B appears n 1 times) = [A,A1 = [A,A,B , . . . ,B] ( B appears n 1 times) =< [h,g](hE H > =x = [x,g , . . . ,g] (g appears n 2 1 times) = a-'ha =(A@A)/
>
>
x,
def (G) AoB AIB [A7B , 01 [A,B7 nl 1-49A, B ; 01 [ A ,A, B ; nl [a791 [ x ,g; 01 15, g; nl ha A#A
>
>
Notation
S#S
M I I , M12, M22, M23, M24 R(q) J1
Sz((r) ZPCQ
r(G)
H9
zLm)
= (s/sl)#(s/sl> the Mathieu groups the Ree group the Janko's first group the Suzuki group the pn-th complex roots of unity, with n running over all integers 0 the minimal number of relations on d(G) generators
>
= g-l H g
direct product of m copies of
En
Rings and Modules
R-homomorphisms of V into W the F-dimension of V the opposite ring of V the center of R tensor product the characteristic of R direct sum direct product the Jacobson radical of R the left socle of R the algebra of R-endomorphisms of V the principal ideal generated by r n x n matrices with entries in R =L@JFV =LBFA the annihilator of V conjugate of V the socle of V the radical of V the commutator subspace of A the lattice of submodules of V the left regular R-module
RR C(V>
A ~ B
T(V)
v*
eij
vn
f 8s
v* f*
RG
SUPP
< Suppx > aug : R G + R aag(x) I ( R G ) or I ( G ) X+ Gn(l+I) FaG [(X) r(X)
vG
R[Xl i(V, W ) P(V> dimFV i n f (V> ext(V)
xv
xG i n f (x) deg( P)
P* V#W VH pG ( C ,d)
the right regular R-module the Clifford algebra of V the graded tensor product the tensor algebra of V the contragredient of V the composition length of V matrix unit direct product of n copies of V tensor product of maps = HomR(V, R ) dual homomorphism group algebra of G over R support of x supporting subgroup of x augmentation map augmentation of x augmentation ideal of R G the sum of all elements of X in R G = {g E Glg - 1 E I} the twisted group algebra of G over F left annihilator of X right annihilator of X induced module polynomial ring on X over R the intertwining number projective cover of V dimension of V over F inflated module extension of V the character afforded by V the induced character the inflated character degree of p contragredient representation outer tensor product of V and W restriction to H induced representation chain complex
Notation
Z ( C ) = Iierd B ( C ) = Imd H(C) zn ( C ) Bn (C) Hn(C) w : H(Ct') + H (C') LnF RnF Ext;(V, -) Exti(-, W ) T O ~ ; ( V-), TOT;(-, W)
cycles boundaries homology module of C = Kerd, = Imdn+l = Zn(C)/Bn(C) connecting homomorphism n-th left derived functor of F n-th right derived functor of F = RnHomA(V,-) = RnHomA(- ,W ) = Ln(V @ A -)
= Ln(-
@A
W)
Field Theory F/ I i ( F :Ii) Gal(F/I i ) K(S) Ii-(al,.. . ,a n )
field extension degree of F over Ii Galois group of F over I< the smallest subfield containing S and li the smallest subfield containing li and
al,-..,an
the trace map the norm of p over F the algebraic closure of F finite field of q elements
Cohomology Theory = Extk,(R, V ) = Hn(ZG,V ) derivations of G into V inner derivations of G into V
I~~G(V) Cn(G, v ) Zn(G7V) Bn(G,V ) 6f
Res Inf Cg
COT Tra H2(G,Alp A : Hom(G, C) t H2(G,A) M(G) M(G)P M(G)[P] M(G)pl M(H)" ~xP(M(G)) M (G)
invariant elements of V standard n-cochains standard n-cocycles standard n-coboundaries coboundary restriction map inflation map conjugation map corestriction map transgression map the p-component of H2(G,A) connecting homomorphism the Schur multiplier of G the pcomponent of M(G) the subgroup of M(G) consisting of all elements whose order divide p
the p'-component of M(G) the K-stable subgroup of M ( H ) the exponent of M(G) the subgroup of M(G) of elements of M(G) of order coprime to n e : H ~ ( GA) , t Hom(M(G),A) the evaluation map S = e, : M(G) -t A the differential of c the cohomology class of f E Z2(G,A) = { f E H2(G,A)l f is symmetric ) Ext(G, A) the second integral homology group of G Z)
f
the natural numbers the rational integers the p-adic integers the rational numbers the real numbers the complex numbers the integers mod m
Set Theory
C
c 1x1
X-Y E
U~EIA~ ni~~Ai
Imf
proper inclusion inclusion the cardinality of the set X the complement of Y in X membership sign the union of the sets the intersection of the sets the image of the map f : X -t Y
Number Theory
a divides b a does not divide b the greatest common divisor of a and b the p p a r t of n the p'-part of n
Group Theory
N a G CG(X) NG(X) o(g) L7.
vl
the subgroup generated by X the cyclic group of order n direct product of G1 and Gz the subgroup of A generated by elements of order dividing p the subgroup of A generated by elements up, a E A N is a normal subgroup of G the centralizer of X in G the normalizer of X in G the order of g E G - 2-1 y -1 xy (unless explicitly stated otherwise) = [G, GI the commutator subgroup of G the center of G the group of all nonsingular linear transformations of V the subgroup of GL(V) consisting of elements of determinant 1 = GL(V)/F' - lv the group of all n x n invertible matrices over R = { A E GLn(R)ldet(A) = 1) = GLn (R)/Z(GLn (R)) = SLn(R)/Z(SLn(R))
Notation
G1@G2
x G(x) Aut(X) OP(G) OPl(G) 0, (GI o n 1 ...?Ti(G)/OT1...T,-l (G) AP g
A[nl
GYP) An sn
t(A) F* U(R)
AH @i€~Gi n i €Gi ~
(G : H) Hom(G, A) P(G1, G2, A ) P(G, A) GAG Kerp
= GLn (Fq ) = SLm(Fq) = PGLn(Fq) = PSLn(Fq) the uni triangular linear group the diagonal subgroup of GL,(q) the orthogonal group of V the special orthogonal group of V the pin group of V the spin group of V the symplectic group the n-th term of the lower central series of G the tensor product of groups GI and G2 the image of x under the action of g E G the stabilizer of x the automorphism group of X the maximal normal psubgroup of G the maximal normal pl-subgroup of G the maximal normal T-subgroup of G = OiTi (G/OTI ...Ti-] (G)) the p-component of A = {a E Alan = 1) p-commutator subgroup of G the alternating group of degrees n the symmetric group of degree n the torsion subgroup of A the multiplicative group of a field F the unit group of R = {a E Alha = a for all h E H ) direct sum of groups G; direct product of groups G; the index of H in G the group of all homomorphisms of G into A the group of all pairings of GI and G2 into A the group of all pairings of G into A = ( G @ G ) / < g @ g l gE G
the kernel of p
>
the a-covering group of G the socle of G the automorphism group of G the Frattini subgroup of G the free group freely generated by X the n-th term of the lower central series of G the dihedral group of order 2m, m 2 the generalized quaternion group of order 4m, m 2 the semidihedral group of order 2m, m >3 =< a , blapm-I = 1, bP = 1, bab-1 = a l + ~ " - 2> the quadratic group the Thompson subgroup of P the subgroup of anti-symmetric pairings the exponent of G the n-th dimension subgroup of G the minimal number of generators of G =< [x,y]lx € y € Y > = [ [ X I , . . ,xn-~],xn] = [ [ X I- ,.. 7 Xn-l],Xn] the free product of A and B the group of universal commutator relations the deficiency of G the second nilpotent product of A and B the wreath product of A and B =A = [A,B , . ..,B] ( B appears n 1 times) = [A,A1 = [A,A,B , . . . ,B] ( B appears n 1 times) =< [h,g](hE H > =x = [x,g , . . . ,g] (g appears n 2 1 times) = a-'ha =(A@A)/
>
>
x,
def (G) AoB AIB [A7B , 01 [A,B7 nl 1-49A, B ; 01 [ A ,A, B ; nl [a791 [ x ,g; 01 15, g; nl ha A#A
>
>
Notation
S#S
M I I , M12, M22, M23, M24 R(q) J1
Sz((r) ZPCQ
r(G)
H9
zLm)
= (s/sl)#(s/sl> the Mathieu groups the Ree group the Janko's first group the Suzuki group the pn-th complex roots of unity, with n running over all integers 0 the minimal number of relations on d(G) generators
>
= g-l H g
direct product of m copies of
En
Rings and Modules
R-homomorphisms of V into W the F-dimension of V the opposite ring of V the center of R tensor product the characteristic of R direct sum direct product the Jacobson radical of R the left socle of R the algebra of R-endomorphisms of V the principal ideal generated by r n x n matrices with entries in R =L@JFV =LBFA the annihilator of V conjugate of V the socle of V the radical of V the commutator subspace of A the lattice of submodules of V the left regular R-module
RR C(V>
A ~ B
T(V)
v*
eij
vn
f 8s
v* f*
RG
SUPP
< Suppx > aug : R G + R aag(x) I ( R G ) or I ( G ) X+ Gn(l+I) FaG [(X) r(X)
vG
R[Xl i(V, W ) P(V> dimFV i n f (V> ext(V)
xv
xG i n f (x) deg( P)
P* V#W VH pG ( C ,d)
the right regular R-module the Clifford algebra of V the graded tensor product the tensor algebra of V the contragredient of V the composition length of V matrix unit direct product of n copies of V tensor product of maps = HomR(V, R ) dual homomorphism group algebra of G over R support of x supporting subgroup of x augmentation map augmentation of x augmentation ideal of R G the sum of all elements of X in R G = {g E Glg - 1 E I} the twisted group algebra of G over F left annihilator of X right annihilator of X induced module polynomial ring on X over R the intertwining number projective cover of V dimension of V over F inflated module extension of V the character afforded by V the induced character the inflated character degree of p contragredient representation outer tensor product of V and W restriction to H induced representation chain complex
Notation
Z ( C ) = Iierd B ( C ) = Imd H(C) zn ( C ) Bn (C) Hn(C) w : H(Ct') + H (C') LnF RnF Ext;(V, -) Exti(-, W ) T O ~ ; ( V-), TOT;(-, W)
cycles boundaries homology module of C = Kerd, = Imdn+l = Zn(C)/Bn(C) connecting homomorphism n-th left derived functor of F n-th right derived functor of F = RnHomA(V,-) = RnHomA(- ,W ) = Ln(V @ A -)
= Ln(-
@A
W)
Field Theory F/ I i ( F :Ii) Gal(F/I i ) K(S) Ii-(al,.. . ,a n )
field extension degree of F over Ii Galois group of F over I< the smallest subfield containing S and li the smallest subfield containing li and
al,-..,an
the trace map the norm of p over F the algebraic closure of F finite field of q elements
Cohomology Theory = Extk,(R, V ) = Hn(ZG,V ) derivations of G into V inner derivations of G into V
I~~G(V) Cn(G, v ) Zn(G7V) Bn(G,V ) 6f
Res Inf Cg
COT Tra H2(G,Alp A : Hom(G, C) t H2(G,A) M(G) M(G)P M(G)[P] M(G)pl M(H)" ~xP(M(G)) M (G)
invariant elements of V standard n-cochains standard n-cocycles standard n-coboundaries coboundary restriction map inflation map conjugation map corestriction map transgression map the p-component of H2(G,A) connecting homomorphism the Schur multiplier of G the pcomponent of M(G) the subgroup of M(G) consisting of all elements whose order divide p
the p'-component of M(G) the K-stable subgroup of M ( H ) the exponent of M(G) the subgroup of M(G) of elements of M(G) of order coprime to n e : H ~ ( GA) , t Hom(M(G),A) the evaluation map S = e, : M(G) -t A the differential of c the cohomology class of f E Z2(G,A) = { f E H2(G,A)l f is symmetric ) Ext(G, A) the second integral homology group of G Z)
f