Abstract Algebra

Sets
Functions
Permutations
1. X={1,2,...,n} & Sn={a:X->X, a bijective}
GROUPS
1. Monoids
  1. Pair(M,*)
  2. M Closed under *(Well Defined)
    1. Not Well Defined=Not Binary Function=Not Closed
  3. e*a=a*e=a
  4. a*(b*c)=(a*b)*c
  5. (Z,-) & (N,exp) NOT monoids
  6. a*b=b*a
2. Pair(G,*)
  1. G Closed under *
  2. e*a=a*e=a
  3. a*(b*c)=(a*b)*c
  4. a*a'=a'*a=e
  5. IF a*b=b*a, => (G,*) = ->Commutative ->Abelian
3. Subgroups
  1. Group (G,*)...Subset H c G...If (H,*) also group, H is Subgroup of G<=>a*b'eH, all a,beH
4. Generators
  1. (M,*) Monoid. Subset A c M "Set Of Generators" of M if : each yeM\{e} can be written using only elements and the operation *
  2. (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 *
    1. <A> = Smallest subgroup of G generated by A containing all elements of A
      1. Groups <A> generated by just one element called Cyclic Groups
GROUP ACTIONS
1. Homomorphism
  1. Monoids (X,$)&(Y,*)...f:(X,$)->(Y,*) is a function f:X->Y such that :f(ex)=ey & f(m$n)=f(m)*f(n) all m,n eX
    1. Monoid Homomorphism f:(X,$)->(Y,*) which is Bijective is called Monoid Isomorphism
  2. 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

Zusammenfassung der Ressource

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