Elements Of Modern Algebra 6Th Edition Chapter 3

Front 1

Group

Back 1

G is closed with respect to *

* Is associative

G has an identity element

Every element has an inverse

Hint 1

Four parts to a group

Front 2

Closed

Back 2

For every x,y that exists in G, x*y exists in G

Front 3

Associative

Back 3

(x*y)*z = x*(y*z) for every x,y,z in G

Front 4

Identity element

Back 4

x*e = e*x = x for every x in G

Front 5

Elements have an inverse

Back 5

For each x in G there exists a y in G such that x*y = y*x = e

Front 6

Abelian

Back 6

A commutative group

Front 7

Commutative

Back 7

x*y = y*x for every x,y in G

Front 8

Property of Group Elements

Back 8

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

Hint 8

5 parts

Front 9

H subgroup of G

Back 9

H ≠ 0 (show there is an identity)

H is closed

H contains inverses

Front 10

Integral expontents

Back 10

a° = e

a¹ = a

a^k+1 = a^k • a

a^-k = (a-¹)^k

Front 11

Cyclic subgroup

Back 11

For any a in G, H = <a> = {aⁿ : n in integers} where H is generated by a

Front 12

a in G where G is a finite group

Back 12

aⁿ = e for some positive integer n

Front 13

|<a>| = m

Back 13

<a> = { e, a, a², ... a^m-1} and m is the expontents (m is the order of a)

Front 14

List generators

Back 14

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]

Hint 14

Relatively prime

Front 15

Find subgroups

Back 15

Ex. Zn, n=8

Positive divisors= 1,2,3,4,8

Z8= <[1]>, <[2]>, <[4]>, <[6]>, <[8]>

(6 comes from multiples of 2)

Front 16

Show a group is cyclic

Back 16

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]}

Front 17

Homomorphism

Back 17

G* and H+ are groups

Ω: G→H such that Ω(g1*g2)= Ωg1 + Ωg2 for every g1,g2 in G (the domain)

Front 18

Isomorphism

Back 18

A homomorphism that is also bijective (1-1 and onto)

Front 19

Epimorphism

Back 19

A surjective homomorphism (onto)

G→G' that is epimorphic implies G' is a homomorphic image of G

Front 20

Kernel

Back 20

Let Ω: G→G' be homomorphic,

Then kerΩ= {g in G| Ω(g) = e'}