Division Algorithm to Infinite Groups

Question 1

Division Algorithm for Z

Answer 1

Let m,n ϵ Z with n>0. Then ∃! integers q,r such that m=nq+r where 0≤r

Question 2

Divisibility: If n|m

Answer 2

then m=nk for some integer k

Question 3

We say that a is congruent to b modulo n, written a≡b(modn) iff___

Answer 3

iff a and b have the same remainder when divided by n or n divides the b - a

Question 4

Proving an Equivalence Relation

Answer 4

1. Non empty
2. reflexive a∼a
3. symmetric if a∼b then b∼a
4. transitive if a∼b and b∼c then a∼c

Question 5

Equivalence Class of a in S under ∼

Answer 5

The equivalence class determined by a is
[a]={x ∈ S | x∼a }

Question 6

T/F. [a]=∅

Answer 6

F because it is of relflexive property

Question 7

An element of an equivalence class is called a _____ of the class

Answer 7

representative

Question 8

A ____ of a set S is a collection of non-empty disjoint subsets of S whose union is S

Answer 8

Partition

Question 9

The equivalence relation “congruence modulo n” partitions Z into n classes ([0],[1],[2],[3],…,[n-1]). The classes are referred to as _____

Answer 9

residue classes modulo n

Question 10

A _____ on a non-empty set S is a rule that assigns to each ordered pair (a,b) of elements of S some elements of S given by (a*b).

Answer 10

binary operation *

Question 11

• is a function from ____ to ____

Answer 11

S x S into S

Question 12

A binary operation on S is said to be ____ on S

Answer 12

closed

Question 13

_____ describes the structure of a FINITE group by arranging ALL the POSSIBLE products of all the group’s elements in a square table

Answer 13

Cayley Table

Question 14

Axioms to be satisfied for Groups

Answer 14

1. G is non-empty
2. binary operation * is associative
3. identity element
4. inverse element

Question 15

T/F. Composition of Functions is not associative.

Answer 15

F

Question 16

Proving a Bjection from D to R

Answer 16

f is well defined
f is one-to-one: f(x)=f(y) then x=y
f is onto: y of R = form for some x of D, then y = f(x)

Question 17

a b w x
[ ] [ ]
c d y z

Answer 17

aw+by ax+bz
[ ]
cw+dy cx+dz

Question 18

General Linear Group of Degree 2 GL(2,R) is a group under _____ with identity _____

Answer 18

matrix multiplication,
1 0
0 1

Question 19

U4={?} is an abelian group under multiplication

Answer 19

U4={z ∈ C| z^4 = 1}

Question 20

A group is abelian iff____

Answer 20

binary operation * is commutative

Question 21

Why is the set of complex numbers under multiplication not a group?

Answer 21

0 has no inverse

Question 22

A function from X to Y is relation where every x in X is mapped to a _____ y in Y.

Answer 22

unique

Question 23

T/F. Composition of Functions is associative.

Answer 23

T

Question 24

What is the general linear group of degree 2 and its binary operation?

Answer 24

GL(2,R) that is G = {[a b….c d]| a,b,c,d ∈ R, ad-bc =/= 0} under matrix multiplication

Question 25

What is Usub4? It is a group under ____.

Answer 25

U4={z ∈ C | z^4 = 1}. Mulitplication for C

Question 26

G is an _____ if G is of infinite order that is G has infinite number of elements.

Answer 26

infinite group

Question 27

T/F. In general Usubn is an abelian group of order n.

Answer 27

T

Question 28

The group U(n) is defined as ___

Answer 28

U(n) = {x ∈ N| x < n and gcd(x,n)=1}

Question 29

The group units of Zsubn is an ABELAIN GROUP formed by _____ under _____

Answer 29

U(n) under multiplication modulo n

Question 30

What is Dsub4?

Answer 30

is the set of rigid motions of a square.

Question 31

Dsub4 is a group under *. Define *.

Answer 31

Define * as a*b, a is performed then b, where a,b are ∈ of Dsub4