Abstract Algebra

Description

College Abstract Algebra Mind Map on Abstract Algebra, created by danny.cashin on 04/19/2014.
danny.cashin
Mind Map by danny.cashin, updated more than 1 year ago
danny.cashin
Created by danny.cashin almost 11 years ago
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Resource summary

Abstract Algebra
  1. Sets
    1. Functions
      1. Permutations
        1. X={1,2,...,n} & Sn={a:X->X, a bijective}
          1. GROUPS
            1. Monoids
              1. Pair(M,*)
                1. M Closed under *(Well Defined)
                  1. Not Well Defined=Not Binary Function=Not Closed
                  2. e*a=a*e=a
                    1. a*(b*c)=(a*b)*c
                      1. (Z,-) & (N,exp) NOT monoids
                        1. a*b=b*a
                        2. Pair(G,*)
                          1. G Closed under *
                            1. e*a=a*e=a
                              1. a*(b*c)=(a*b)*c
                                1. a*a'=a'*a=e
                                  1. IF a*b=b*a, => (G,*) = ->Commutative ->Abelian
                                  2. Subgroups
                                    1. Group (G,*)...Subset H c G...If (H,*) also group, H is Subgroup of G<=>a*b'eH, all a,beH
                                    2. 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 *
                                        1. (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
                                      2. 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
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