SEM1 1st half Flashcards
Define an isometry
Function Q: X to X is an isometry if:
Q is bijective
for all x,y in X the distance between x and y is equal to the distance between Q(x) and Q(y)
Give examples of Transformations that are isometries
Also give a property about the invertability of isometric functions
Translation, Reflection, Rotation
All isometries are Invertible and the inverse is also an isometry
Given 2 isometries from X to X, f and g, is the composition of f and g an isometry.
If so then prove it
By the definition of isomtery g o f can be seen to be bijective.
as f is isometric the distance between x,y is equal to the distance between f(x) and f(y) which are both contained within X therefore applying g to f(x),f(y) the distance stays preserved due to the isometric property of g.
List the Group Axioms
G is a non empty set
binary operation is associative
there exists an identity element
each g in G has an inverse
Briefly explain why the pair (Z,+) is a group
G0: 0 is contained within Z, non empty
G1: if a,b are contained within Z a+b is in Z, and (a+b)+c=a+(b+c), associative
G2: The element 0 is the identity element
G3: -a is contained within Z s.t. a+(-a)=0
Define groups G and H to be isomorphic
an isomorphism from G to H is a bijective function such that: for all a,b in G we have phi(ab)=phi(a)phi(b)
Define a subgroup H of G
for all g,h in H we have gh in H
(H,) is a group
prove that the identity element of G must be equal to the identity element of H, if H is a subgroup of G
lemma 2.1.4 in notes
definition of a cyclic sub group generated by a
if there exists an element a in G such that all elements in G are powers of a
describe SL(n,k)
the special linear groups, e.g. nxn matricies with determinant 1
definition of a group homomorphism
for all a,b in G we have phi(ab)=phi(a)phi(b)
4 properties of homomorphisms
phi maps the identity element of G to the identity element of H
for all a in G phi(a^-1) = (phi(a))^-1
The image of phi is a subgroup of H: Im(phi) < H
The kernel of phi is a subgroup of G: Ker(phi)<G
define order of a group
the order |G| of a group G is the cardinaility of the set G
define the order of an element in G
the order of element a is the smallest positive integer m s.t. a^m = e
if m does not exist the element a has infinite order
4 properties of a in G where |a|=n
elements a,a^2,…,a^n=e are all distinct
if s is an integer with s=kn+r a^s=a^r
if s,t are in Z then a^s=a^t implies s=tmod(n)
a^s=e implies s is divisible by n
