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20 Cards in this Set
- Front
- Back
- 3rd side (hint)
Group |
G is closed with respect to * * Is associative G has an identity element Every element has an inverse |
Four parts to a group |
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Closed |
For every x,y that exists in G, x*y exists in G |
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Associative |
(x*y)*z = x*(y*z) for every x,y,z in G |
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Identity element |
x*e = e*x = x for every x in G |
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Elements have an inverse |
For each x in G there exists a y in G such that x*y = y*x = e |
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Abelian |
A commutative group |
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Commutative |
x*y = y*x for every x,y in G |
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Property of Group Elements |
Identity is unique For each x in G, x–¹ in G is also unique For x in G, (x-¹)-¹ = x y,x in G, (xy)-¹ = y-¹x-¹ (reverse order if not abelian) a,x,y in G, ax = ay implies x=y |
5 parts |
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H subgroup of G |
H ≠ 0 (show there is an identity) H is closed H contains inverses |
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Integral expontents |
a° = e a¹ = a a^k+1 = a^k • a a^-k = (a-¹)^k |
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Cyclic subgroup |
For any a in G, H = <a> = {aⁿ : n in integers} where H is generated by a |
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a in G where G is a finite group |
aⁿ = e for some positive integer n |
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|<a>| = m |
<a> = { e, a, a², ... a^m-1} and m is the expontents (m is the order of a) |
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List generators |
Ex. G=<a> is a cyclic group of order 24 Generators of G: a, a^5, a^7, a¹¹, a¹³, a^17, a^19, a²³
Ex. Zn, n=12 Generators: [1], [5], [7], [11] |
Relatively prime |
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Find subgroups |
Ex. Zn, n=8 Positive divisors= 1,2,3,4,8 Z8= <[1]>, <[2]>, <[4]>, <[6]>, <[8]> (6 comes from multiples of 2) |
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Show a group is cyclic |
Ex. Z5 Z*5= {[1],[2],[3],[4]} (find the one with powers respect to 5) 2°=1, 2¹= 2, 2²= 4, 2³=8 which cycles to 3 <[2]>= {[1],[2],[4],[3]} |
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Homomorphism |
G* and H+ are groups Ω: G→H such that Ω(g1*g2)= Ωg1 + Ωg2 for every g1,g2 in G (the domain) |
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Isomorphism |
A homomorphism that is also bijective (1-1 and onto) |
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Epimorphism |
A surjective homomorphism (onto) G→G' that is epimorphic implies G' is a homomorphic image of G |
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Kernel |
Let Ω: G→G' be homomorphic, Then kerΩ= {g in G| Ω(g) = e'} |
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