Abstract Algebra

Abstract Algebra

1. Sets

2. Functions

3. Permutations

   • X={1,2,...,n} & Sn={a:X->X, a bijective}

4. GROUPS

   • Monoids
     • Pair(M,*)
     • M Closed under *(Well Defined)
       • Not Well Defined=Not Binary Function=Not Closed
     • e*a=a*e=a
     • a*(b*c)=(a*b)*c
     • (Z,-) & (N,exp) NOT monoids
     • a*b=b*a
   • Pair(G,*)
     • G Closed under *
     • e*a=a*e=a
     • a*(b*c)=(a*b)*c
     • a*a'=a'*a=e
     • IF a*b=b*a, => (G,*) = ->Commutative ->Abelian
   • Subgroups
     • Group (G,*)...Subset H c G...If (H,*) also group, H is Subgroup of G<=>a*b'eH, all a,beH
   • Generators
     • (M,*) Monoid. Subset A c M "Set Of Generators" of M if : each yeM\{e} can be written using only elements and the operation *
     • (G,*) Group. Subset A c M "Set Of Generators" of G if : each yeG can be written using only elements and their inverses and the operation *
       • <A> = Smallest subgroup of G generated by A containing all elements of A
         • Groups <A> generated by just one element called Cyclic Groups

5. GROUP ACTIONS

   • Homomorphism
     • Monoids (X,$)&(Y,*)...f:(X,$)->(Y,*) is a function f:X->Y such that :f(ex)=ey & f(m$n)=f(m)*f(n) all m,n eX
       • Monoid Homomorphism f:(X,$)->(Y,*) which is Bijective is called Monoid Isomorphism
     • Groups (X,$)&(Y,*)...f:(X,$)->(Y,*) is a function f:X->Y such that :{{f(ex)=ey}} & f(m$n)=f(m)*f(n) all m,n eX