Flashcards in Alg Deck (68):

1

## standard sets

###
1

Z,Q,R,C,M_{mxn}(S), M_n(S)"+" superscript, "+" with overbar, *

2

## binary operation on S

### 2

3

## closed under an operation

### 3

4

## associative

### 4

5

## semigroup

### 5

6

## identity

### 6

7

## monoid

### 7

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## inverse

### 8

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## group

###
9

defn, equivalently 4 conditions

10

## properties of groups

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10

6 of em

11

## commutative

### 11

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## Abelian

### 12

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## order of S

### 13

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## examples of groups

###
14

7 groups, 2 more

15

## general linear group

### 15

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## special linear group

### 16

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## nth roots of unity

### 17

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## symmetric group on n letters

### 18

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## dihedral group

### 19

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## group table

### 20

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## additive group of integers modulo n

### 21

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## multiplicative group of integers modulo p

### 22

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## cyclic group

### 23

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## generator of (Z_n,+)

### 24

25

## when a^m is a generator of G

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## subgroup

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## center

### 27

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## cyclic subgroup generated by a

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## order of an element

### 29

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## finitely generated group

### 30

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## relations

###
31

defn, Klein four-group presentation

32

## Lagrange's theorem

### 32

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## Abelian subgroup of certain order

### 33

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## subgroups of cyclic groups

### 34

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## Cauchy's theorem

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## Sylow's first theorem

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## isomorphic

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## direct product

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## when Z_m+Z_n+... is cyclic

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## elementary divisors of G

### 40

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## invariant factors of G

### 41

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## coset

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## group homomorphism

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## monomorphism

### 44

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## epimorphism

### 45

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## isomorphism

### 46

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## endomorphism

### 47

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## automorphism

### 48

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## properties of homomorphisms

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49

6 of em

50

## kernel

### 50

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## ring

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3 main conditions, total of 8 sub-conditions

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## ring with unity

### 52

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## commutative ring

### 53

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## subring

### 54

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## characteristic of the ring

### 55

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## ring of integers

### 56

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## ring of integers modulo n

### 57

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## ring of Gaussian integers

### 58

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## ring of real-valued functions on R

### 59

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## ring homomorphism

### 60

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## properties of ring homomorphism

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61

3 of em

62

## integral domain

### 62

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## zero divisor

### 63

64

## zero divisors in Z_n

### 64

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## integral domains Z_n

### 65

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## unit

### 66

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## division ring

### 67

68