Intro to Abstract Algebra

Question 1

What is a binary operation on a set S

Answer 1

A rule for where every two elements of S gives another element of S

Question 2

Define R*

Answer 2

The set of reals excluding zero

Question 3

When is a binary operation commutative

Answer 3

When ab=ba for all a,b in S

Question 4

When is a binary operation associative

Answer 4

When a(bc)=(ab)c for all a,b,c in S

Question 5

Define a group and its four properties

Answer 5

a pair (G, o) where G is a set and o is a binary operation such that the four following properties hold

i) closure - if a,b in S, ab is in G
ii) associativity - a(bc)=(ab)c for all a,b,c in G
iii) Existence of an identity
iv) Existence of an inverse

Question 6

Theorem; Let (G, o) be a group. Then (G, o) has a unique identity element

Answer 6

Question 7

Theorem; Let G be a group and let a be an element of G. Then a has a unique inverse

Answer 7

Question 8

Define GL(2)R

Answer 8

Question 9

Theorem; GL(2)R is a group under matrix multiplication

Answer 9

Question 10

What is the order of an element?

Answer 10

The smallest positive integer such that a^n=1. If there is no such positive integer n, we say a has infinite order

Question 11

Lemma; Let G be a group and g be an element of G

i) g has order 1 if and only if g is the identity element
ii) let g^m be a non-zero integer. then g^m=1 if and only if g has finite order d with d|m

Answer 11

Question 12

What is the order of a group G

Answer 12

The number of elements of G

Question 13

What is the relationship of the order of G and an element g in G?

Answer 13

By Lagranges Theorem, the order of g divides the order of G

Question 14

When is a subset H of a group G a subgroup, There are three conditions

Answer 14

1 lies in H

if a,b is in H, then ab is in H

if a is in H, then inverse a is in H

Question 15

Define the Circle group S

Answer 15

Question 16

Lemma; Un (the roots of unity) are a subgroup of C* of order n

Answer 16

Question 17

What is the relationship between the order of H and the order of G, where H is a subgroup of G

Answer 17

The order of H divides the order of G

Question 18

Define SL(2)R

Answer 18

Elements are 2 x 2 matrices where det(A)=1

Question 19

Define the cyclic subgroup generated by an element g and prove that it is a subgroup

Answer 19

Question 20

Theorem; Let G be a group and let g be an element of finite order n. Then <g> = {1, g, .... , g^(n-1)}. In particular, the order of the subgroup <g> is equal to the order of g</g></g>

Answer 20

Question 21

Define the left coset of a subgroup H and a right coset of subgroup H

Answer 21

Left; gH = {gh ; h in H}

Right; Hg = {hg ; h in H}

Question 22

Define the index of H in G

Answer 22

The index of a subgroup H is the number of left cosets of H in G. Written as [G : H]

Question 23

Lemma; Let G be a group and H a finite subgroup. If g in G then gH and Hg have the same number of elements as H

Answer 23

Question 24

Lemma; Let G be a group and H a subgroup. Let g(1), g(2) be elements of G. Then the cosets g(1)H and g(2)H are equal or disjoint

Answer 24

Question 25

What is the relationship between the index of the subgroup H in G and the number of elements in H and G (Lagranges Theorem). Prove

Answer 25

Question 26

Let H be a subgroup, and a,b elements of G. When are a and b congruent modulo H

Answer 26

When a-b is in H. We write a= b modH

Question 27

Define the Congruence Class of a modulo H

Answer 27

a = {b in G; b = a mod H}

Question 28

Lemma; Let (G,+) be an additive abelian group and H a subgroup. Take a in G, and let a(hat) be the congruence class of a modulo H. Then a(hat) = a+H

Answer 28

Question 29

Let m >=2 be an integer. Let a, a’, b, b’ be satisfying a = a’ mod Z/mZ and b = b’ mod Z/mZ. Then ab = a’b’ mod Z/mZ

Answer 29

Question 30

Define Sym(A)

Answer 30

Let A be a set and f,g be functions from A to A. Sym(A) is the set of bijections from A to A

Question 31

Define the domain and co-domain of a function f that maps A to B

Answer 31

Domain - A

Codomain - B

Question 32

State the Pigeon-Hole Principle

Answer 32

Let A be a finite set. Then a function f that maps A to A is injective if and only if it is surjective

Question 33

Theorem; Let A be a set. Then Sym(A) is a group with Id(A) as the indentity element

Answer 33

Question 34

What is the Set Sn

Answer 34

The group Sym{1,2,…..,n}, the nth symmetric group

Question 35

Theorem; Sn has order n!

Answer 35

Question 36

Lemma; Disjoint Cycles Commute

Answer 36

Question 37

What is a transposition. Write (a1,a2,….,am) as a product of transpositions

Answer 37

A cycle of length 2.

(a1,a2,….,am) = (a1,am)……(a1,a3)(a1,a2)

Question 38

Define Pn, the nth alternating polynomial

Answer 38

Question 39

Lemma; Let T be a transposition in Sn. Then T(Sn) = -Sn

Answer 39

Question 40

Define the nth Alternating Subgroup An

Answer 40

Question 41

Theorem; An is a subgroup of Sn

Answer 41

Question 42

Theorem; Let n>1, then An has order equal to 1/2 Sn = n!/2

Answer 42

Question 43

Define a Ring

Answer 43

R is a set and +, * are binary operations on the set such that there

is

i) closure
ii) associativity of addition and multiplication
iii) existence of additive inverse, multiplication identity
iv) distributivity

Question 44

Define a Subring and its 4 conditions

Answer 44

S, a subset of R, is a subring of R if S is a ring with respect to the same operations.

i) 0 ,1 in S
ii) a, b in S then ab is in S
iii) a,b in S then a + b is in S
iv) if a is in S then -a is in S

Question 45

What is the easiest way to prove a set is a Ring

Answer 45

Prove it is a subring of a well known ring

Question 46

Define a unit of a ring

Answer 46

Let R be a ring. An element u is a unit if there is an element v in R such that uv=1. ie it has a multiplicative inverse

Question 47

What is a field?

Answer 47

A Ring such that every non-zero element is a unit

Question 48

Theorem; Let a be in Z/mZ. Then a is a unit in Z/mZ if and only if gcd(a,m)=1. Thus

(Z/mZ)* = { a; 0

Answer 48

Question 49

When is Z/mZ a field

Answer 49

m is prime

Question 51

What are the gaussian integers Z[i]

Answer 51

a + bi

a,b in Z