Division Algorithm to Infinite Groups Flashcards
Division Algorithm for Z
Let m,n ϵ Z with n>0. Then ∃! integers q,r such that m=nq+r where 0≤r
Divisibility: If n|m
then m=nk for some integer k
We say that a is congruent to b modulo n, written a≡b(modn) iff___
iff a and b have the same remainder when divided by n or n divides the b - a
Proving an Equivalence Relation
- Non empty
- reflexive a∼a
- symmetric if a∼b then b∼a
- transitive if a∼b and b∼c then a∼c
Equivalence Class of a in S under ∼
The equivalence class determined by a is
[a]={x ∈ S | x∼a }T/F. [a]=∅
F because it is of relflexive property
An element of an equivalence class is called a _____ of the class
representative
A ____ of a set S is a collection of non-empty disjoint subsets of S whose union is S
Partition
The equivalence relation “congruence modulo n” partitions Z into n classes ([0],[1],[2],[3],…,[n-1]). The classes are referred to as _____
residue classes modulo n
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).
binary operation *
- is a function from ____ to ____
S x S into S
A binary operation on S is said to be ____ on S
closed
_____ describes the structure of a FINITE group by arranging ALL the POSSIBLE products of all the group’s elements in a square table
Cayley Table
Axioms to be satisfied for Groups
- G is non-empty
- binary operation * is associative
- identity element
- inverse element
T/F. Composition of Functions is not associative.
F
Proving a Bjection from D to R
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)
a b w x
[ ] [ ]
c d y z
aw+by ax+bz
[ ]
cw+dy cx+dz
General Linear Group of Degree 2 GL(2,R) is a group under _____ with identity _____
matrix multiplication,
1 0
0 1
U4={?} is an abelian group under multiplication
U4={z ∈ C| z^4 = 1}
A group is abelian iff____
binary operation * is commutative
Why is the set of complex numbers under multiplication not a group?
0 has no inverse
A function from X to Y is relation where every x in X is mapped to a _____ y in Y.
unique
T/F. Composition of Functions is associative.
T
What is the general linear group of degree 2 and its binary operation?
GL(2,R) that is G = {[a b….c d]| a,b,c,d ∈ R, ad-bc =/= 0} under matrix multiplication
What is Usub4? It is a group under ____.
U4={z ∈ C | z^4 = 1}. Mulitplication for C
G is an _____ if G is of infinite order that is G has infinite number of elements.
infinite group
T/F. In general Usubn is an abelian group of order n.
T
The group U(n) is defined as ___
U(n) = {x ∈ N| x < n and gcd(x,n)=1}
The group units of Zsubn is an ABELAIN GROUP formed by _____ under _____
U(n) under multiplication modulo n
What is Dsub4?
is the set of rigid motions of a square.
Dsub4 is a group under *. Define *.
Define * as a*b, a is performed then b, where a,b are ∈ of Dsub4
