.Open Group Theory . Motivation Groups turn up whenever we have a set of reversible operations. The numbers with adddition, numbers (without zero) and multiplication, the movements of a figure on the plane, or a beer glass in space all follow a similar pattern of operations that we call a group. A group is anything that has most of the algebraic properties of the numbers with respect to either addition or multiplication (but not both). Groups are are a rich and well explored algebra. . Basics monoids::=http://www/dick/maths/math_33_Monoids.html |-(basis): \$monoids. GROUP::=following .Net Set::Sets, operation::Associative(Set), op::=operation, unit::units(Set,op). u:=unit. |-(G0): MONOID(Set, op, u). inverse::Set->Set, i:=inverse, |-(G1): for all x:Set, x op i(x)=u=i(x) op x. .See Properties .Close.Net |-(Group_is_algebra): ALGEBRA(\$GROUP, Group, group). (Group_is_algebra)|- Group::=\$ \$GROUP. (Group_is_algebra)|- group::={Set:Sets || \$GROUP(Set, op, u, i)}, group::={Set:Sets || \$GROUP(Set, operation=>(1st), unit=>(2nd), inverse=>(3rd))}. monoid::Group->Monoid=(Set=>(_).Set, op=>(_).operation, unit=>(_).unit). semigroup::Group->Semigroup=(Set=>(_).Set, op=>(_).operation). congruence::Group->Equivalence_relations=congruence(monoid(_)). . Alternative definition |-(one-sided): Given just left identities and inverse then a monoid must be a group. .Source Georges Pappy's book on Groups and Birkhoff and Bartee 70, page 213. |- Group=\$ \$PAPPY. PAPPY::=following .Net A semigroup in which equations can be solved. |-(PG0): SEMIGROUP(Set, op). u::Set. i::Set---Set. |-(PG1): for all a,b:Set, some x,y :Set (a op x = b = y op a). (PG1)|-(PGu): u= the x (for all a (a op x=a=y op a)). (PG1,PGu)|-(PGi): i= the i( for all a ( a op i(a) =u)). . Proof of PGu .Let |- a:G. (PG1(a=>a, b=>a))|- for some x,y:Set(a op x=a=y op a). ()|- some left_units and some right_units. ()|-one unit. .Close.Let . Proof of PGi .Let |- a:G, (PG1(a=>a, b=>u)) |- inverses ... .Close.Let .Close.Net . Properties .See http://www/dick/maths/math_33_Monoids.html#unique_inverses ()|- (unique unit): For all Group G, one u:G ( for all x:G( x op u = u op x = x)). ()|- (unique inverse): For all Group G, one i:G ( for all x:G( x op i(x)= u )). ()|- (cancelation): For all Group G, for all a:G ( for all x,y(if a op x = a op y then x=y) and for all x,y(if x op a = y op a then x=y). ()|- (equation_solvable): For all G,a,b( for one x( a op x =b) and one x ( x op a =b ) ) . ()|- (solve_equation): For all G,a,b( the x( a op x =b)= i(a) op b and the x ( x op a =b )=b op i(a) ) . ()|-(inv unit): For G, i(u)=u. ()|-(dist inv): For G, i(a op b) = i(b) op i(a). Note, that by the STANDARD defitions (op) is a serial operator, .See http://www/dick/maths/math_11_STANDARD.html ()|-(sum): For G, A,B:Finite_Sets, f:G^(A|B), op(f) = op(f o B) op op(f o B) op i( op(f o (A&B))). . Special groups m_group::=multiplicative_group. multiplicative_group::={Set||\$GROUP(op=>., u=>1, i=>map s(s^-1)) and Net{ For x,y:Set, (x y):=x.y}}. a_group::=additive_group. additive_group::={Set||\$GROUP(op=>+, u=>0, i=>-) and Net{+ in commutative(Set)}}. Abelian_group::=\$ \$GROUP and Net{ op in commutative(Set)} Traditionally commutative groups use additive notation while non_commutative groups use the multiplicative notation: Do properties like (ab)^n=a^n b^n force a group to be abelian? .See Gallian & Reid 93, "Abelian Forcing Sets", Am Math Monthly June-July 93 pp580-582. FREE_GROUP::=Net{ Elements::Finite_sets. Inverse::Elements --- Elements. |- \$GROUP( Set=> #( Elements | Inverses), op=>(!), u=>(), i=>Inverse). |- For all s1,s2:Set, e:Elements, s1 ! e ! Inverse(e) ! s2 = s1! s2. }=::FREE_GROUP. For S:Finite_sets, i:S---S, Free_group(S, i)={Set || FREE_GROUP( Elements=>S, Inverse=>i, ...)}. ()|- Free_group(S,i)=Monoid( S || for all s( s i(s) = i(s) s = u ) ). .Open Multiplicative Groups . Congruency For G:m_group~`numeric`, x,y:G, x^y::=y^-1.x.y For groups that contain numbers x^n is ambiguous - use x^^y for congruency . Commutator For G:m_group, x,y:G, [x,y]::=x^-1.y^-1.x.y .Close Multiplicative Groups . Actions of a Group The isomorphisms of a set form a group. For set S, iso(S)::group= \$GROUP( S---S, (o), Id(S), (map[i](/i)) ). As in \$monoid theory, we can define the actions of a group -- but the actions must be reversible: For M:group, S:set, actions(M,S)::=(group)(M)->iso(S). Most of the results .See ./math_33_Monoids.html#Actions of a Monoid for monoids can be applied to Actions of a Group as well. We can also show some new results like the Burnside Lemma Here is a more traditional presentation of actions of a group action GROUP_ACTION::=following .Net G:: group(o, u, i) = \$given, X:: Sets = \$given, *:: G >< X -> X. |-(GA1): for all x, u * x = x |-(GA2): for all x, g1,g2, (g1 o g2) * x = g1 * ( g2 * x). We say that X is a G-Set. G[x] ::= {g || g*x = x}, X[g] ::= {x || g*x = x}, x1 ~ x2 ::= for some g, g*x1=x2. G x ::= x/~. (above)|- for x, G x \$sub_group G. (above)|- |G x| = |G : G[x]| = |G| / |G[x]|. orbit = {G x || for some x}. (above)|- (burnside_s_lemma) : |orbit| * |G| = +[g:G] (X[g]). .Close.Net GROUP_ACTION . Subgroups For G:\$ \$GROUP, subgroups(G) ::={ H:@G||H in \$ \$GROUP}, For H,G, H sub_group G::= H in subgroups(G). ()|- for H:@G(H sub_group G iff for all x,y:H(x op i(x) in H). ()|- {{u},G}==>sub_groups(G). ()|- (Set=>sub_groups(G),R=>`==>`)in poset. For A:@G, ::=min{H:subgroups(G)||A==>H}. ()|-={op(x)||x:#(A|i(A))} =( H:@G||for all a:A,h:H({h^-1, a op h, h op a}==>H)). normal_subgroups(G)::={H:subgroups(G)||for all x:G(x.H=H.x)}. normal(G) ::=normal_subgroups(G). For X,Y:subgroups(G), X orthogonal Y::@=( X&Y={u} and X op Y =G). . Generators and Relators For S:@G, generated_group(S) ::=the smallest { H : subgroups(G) || S==> H }. .Example DIHEDRAL::=Net{n:Nat. n>2. m_group(#{R, V}). V V=1. R^n=1. R V = V R^-1.} For A:Finite_set, i: A---A, S:@#A , Group(A||S) ::=Free_group(A, i )/R where R=min{E:congruence(#A)||for all s:S(s E 1)} So the Dihedral group G in \$ \$DIHEDRAL is isomorphic to Group({R,V}|| VV, V^-1 R V R^-1, R^n). . Morphisms For G1,G2:\$ \$GROUP, a:arrow, (group)G1 a G2::=(monoid) Monoid(G1) a Monoid(G2)& {f || f o i=i o f}. .Source Birkhoff & Bartee 70 ()|- For G1,G2:\$ \$GROUP, a:arrow, (group)G1 a G2 ::= (semigroup) Semigroup(G1) a Semigroup(G2). .Note However a semigroup morphism between monoids may not be a monoid morphism. . Left Cosets. For G:m_group, H: sub_groups(G), x,y:G, (x = y wrt H) ::=(x.H=y.H), (=wrt H) ::=rel [x,y](x=y wrt H), for x:G, x/H::=x/(=wrt H), left_cosets(H,G) ::=G/H. |-(LC0): For G,H, (=wrt H) in congruence(monoid(G)). COSETS::=Net{ G::m_group, H::sub_groups(G). (def)|-(C1): x/H={y || y.H=x.H}, (def) |-(C2): left_cosets(H,G)= G/(=wrt H). (def)|-(C3): For x,y,H, x=y wrt H iff y^-1.x in H ()|-(C4) for all x:G (x.H=x/H). . Proof of C4 Let{ x:G, (C4.0): x/H={y||y.H=x.H}, ()|-(C4.1): ( for all h,some g(y=x g h^-1) and for all h, some g(y=x h g^-1) iff for some h(y=x h). LHS::= for all h,some g(y=x g h^-1)&for all h, some g(y=x h g^-1), RHS::=for some h(y=x h). ()|-(C4.2): if LHS then RHS. ()|-(C4.3): if RHS then LHS. x/H= {y||for some h(y=xh)} }. . Proof of C4.2 Let{ |- (C4.2.0): LHS, |- (C4.2.1):not RHS. (C4.2.1)|- (C4.2.2): for all h(y<>x h). (C4.2.0)|- (C4.2.3): for all h, some g(y=x g h^-1). (C4.2.0)|- (C4.2.4): for all h, some g(y=x h g^-1). (C4.2.3, h:=h0, g:=g0) |-(C4.2.5): y=x g0 h0^-1 . (C4.2.2, h:=g0 h0^-1) |- (C4.2.6): y<>x g0 h0^-1 . (C4.2.5, C4.2.6)|- y <> y. }. . Proof of C4.3 Let{ |-(C4.3.0): not LHS, |-(C4.3.1): RHS, ()|-(C4.3.2):y=x h0. ()|-(C4.3.3): for some h,all g(y<>x g h^-1) or for some h, all g(y<>h g^-1) ()|-(C4.3.4): all g(y<>x g h1^-1) or all g(y<>x g h1^-1) ()|-(C4.3.5): either all g(y<>x h1 g^-1) or all g(y<>x h1 g^-1) ()|-(C4.3.6): y<>x h0 h1 h1^-1 ()|-(C4.3.7): g=h1 h0 }. ()|-(C4.4):for all x:G(x.H in G/H). ()|-(C4.5): for all x:G (A;(x._) in A---x.A). ()|-(C4.6):for all x:G, (x.A);(x^-1._) = / (x._). ()|-(C4.7): for all A:G/H( A---H ). ()|- (C4.8): If H in normal(G), map[x](x/H) in (group)G>==G/H }=::COSETS. ()|- For all G,H(COSETS). . Kernels Cosets and morphisms As with monoids the possible morphisms from a group into another group are reflected in the normal subgroups. For any morphism f:(group)G1>->G2 we can define a normal subgroup of G1 called the kernel of f: ker(f) ==> G1 >==coi(f) --- img(f) ==> G2. Given a normal subgroup N of G1 then we can define a group morphism G1>==G1/N. . Burnside's Theorem ()|-(burnside): For G:m_group, H:normal(G), |G| = |G/H| * |H|. ()|-For G1,G2:group, f:(group)G1->G2, ker(f) in normal(G1) and G1/ker(f)=coi(f) ()|-for all G1,G2:group, f:(group)G1>--G2, |G1|=|G2|*|ker(f)| ()|-For H,K:normal(G), if H==>K then G/H>--G/K |-For G,X:group, x:(group)G->X, some Y:group, y:(group)G->Y ( G---\$ Net{x:X y:Y}and ker(x)orthogonal ker(y) ). . Direct Sum ()|-For H,K:sub_group(G), G---H>G2, (h o (_)) in G1^S(group)->G2^S. For G1,G2:group, G1> G1.Set> map x,y:Set((x(1) G1.op y(1),x(2) G2.op y(2))), u=> (G1.u, G2.u), i=> map x((G1.i(x(1)),G2.i(x(2)))) ). . Bilinear Maps in Abelian Groups For A,B,C:a_group,(bilinear) A>C::={f^C(A>C) and for all b:B((f((_),b)) in (group) A->C). . Groups of permutations of a set For S:Set, permutations(S) ::={G:operators(S)||for g in G(g:S---S)} For A are G, T are S, A.T={g.s||g in A and s in T}, G/s::={g || g.s=s}, S/g::={s || g.s=s}, Orbit(T) ::={G.{s} || s in T}, Isotropy(G,s) ::={/s || s in S, invariant(T)(G) ::={T/g || g in G}, s1=s2 wrt G ::= for some g(x=g y). s1(= mod G)s2::=s1=s2 wrt G. ()|-(GP0): for all s(G/s in m_group). ()|-(GP1): for all s(G/s(group)==>G). ()|-(GP2): (=mod G) in equivalence_relation (burnside) |-(GP3): |S/(=mod G)| = +[g:G](|S/g|)/|G| `Average orbit` For S:Set, G:permutations(S), base(S,G) ::={B:@S|| G.B=S and no (G~{1}).B &B}. for all s:S, one (b,g):B>>B, B' not_in base, ? ?? for all B'<<=B, B' not_in base(s,G), ? for all B1,B2:Base(S,G)( B1---B2), S---B>G For S:Set, G:permutations(S), s:S, x,y:G, x=y wrt s ::= x.s=y.s. |-For S:Set, G:permutations(S), s:S, x,y:G, .( x=y wrt s in equivalence_relation(G) and x=y wrt G/S iff x=y wrt s .) .Hole .Open Permutation Groups and the Symmetric Groups For n:Nat, Symmetric_group(n)::=(Set=>(1..n---1..n), op=>o, u=>I, i=>(/)). For n:Nat, S(n) ::=Symmetric_group(n). Use multiplicative notation . Cycle notation A permutation of a finite set can be described as a product of cycles. A Cycle is a list of elements where the permuation changes each element into the next one. For example cycle(1,2,3) maps 1 to 2, 2 to 3, and 3 to 1. For Nat m1..n, cycle(l)::= the p:S(n) ( for i:1..m-1(p(l(i))=l(i+1)) and p(l(m))=l(1) and for s:S(n)~img (l)(p(s)=s)). Cycles are independent if they have no common elements: no(img l1 & img l2). Independent cycles commute. .Hole for f:Nat, f_cycle(p)::={q || for some m:Nat0(q=f^n(p))}, f_period(p)=min{m:Nat0||f^m(p)=p} . Cannonical decomposition The order of a set of disjoint cycles does not matter and there is a cannonical order that can be defined. For f:S[n], f=*[i:1..l](cycle(l(i))) and for all iS[n]) . Proof of laplace Let { |-n=|G|, ()|- some c: G---1..n.(c:G---1..n) ()|- c in G---1..n and /c in 1..n---G and c;/c=Id(G) and /c;c=Id(1..n). e:=map [x:G](/c;(x._);c). ()|- (1): for each a, e(x) in S[n], ()|- (2): e(1)=S[n].unit. ()|- (3): e(a.b)=e(a) e(b). (1,2,3) |- (4): e in (group)G->S[n] ()|- (5): for a<>b, e(a)<> e(b). (5) |- e in (group)G-->S[n] } .Close Permutation Groups and the Symmetric Groups . Geometries, pattern and symmetry Studying the symmetry groups of objects in space is a way of studying space itself. It is also put to use in chemistry and physics wherein the categorisation of all possible symetries in 3 and 4 dimensional space has some significant implications. M C Escher's research into the symetry's of 2 dimensional space is the basis of much of his art. . Supplements and factors .Hole .Close Group Theory