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768813
Abstract Algebra
Description
College Abstract Algebra Mind Map on Abstract Algebra, created by danny.cashin on 04/19/2014.
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abstract algebra
college
Mind Map by
danny.cashin
, updated more than 1 year ago
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Created by
danny.cashin
almost 11 years ago
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Resource summary
Abstract Algebra
Sets
Functions
Permutations
X={1,2,...,n} & Sn={a:X->X, a bijective}
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
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
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