Intro to Abstract Algebra Flashcards
What is a binary operation on a set S
A rule for where every two elements of S gives another element of S
Define R*
The set of reals excluding zero
When is a binary operation commutative
When ab=ba for all a,b in S
When is a binary operation associative
When a(bc)=(ab)c for all a,b,c in S
Define a group and its four properties
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
Theorem; Let (G, o) be a group. Then (G, o) has a unique identity element
Theorem; Let G be a group and let a be an element of G. Then a has a unique inverse
Define GL(2)R
Theorem; GL(2)R is a group under matrix multiplication
What is the order of an element?
The smallest positive integer such that a^n=1. If there is no such positive integer n, we say a has infinite order
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
What is the order of a group G
The number of elements of G
What is the relationship of the order of G and an element g in G?
By Lagranges Theorem, the order of g divides the order of G
When is a subset H of a group G a subgroup, There are three conditions
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
Define the Circle group S
Lemma; Un (the roots of unity) are a subgroup of C* of order n
What is the relationship between the order of H and the order of G, where H is a subgroup of G
The order of H divides the order of G
Define SL(2)R
Elements are 2 x 2 matrices where det(A)=1
Define the cyclic subgroup generated by an element g and prove that it is a subgroup
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>
Define the left coset of a subgroup H and a right coset of subgroup H
Left; gH = {gh ; h in H}
Right; Hg = {hg ; h in H}
Define the index of H in G
The index of a subgroup H is the number of left cosets of H in G. Written as [G : H]
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
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
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
Let H be a subgroup, and a,b elements of G. When are a and b congruent modulo H
When a-b is in H. We write a= b modH
Define the Congruence Class of a modulo H
a = {b in G; b = a mod H}
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
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
Define Sym(A)
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
Define the domain and co-domain of a function f that maps A to B
Domain - A
Codomain - B
State the Pigeon-Hole Principle
Let A be a finite set. Then a function f that maps A to A is injective if and only if it is surjective
Theorem; Let A be a set. Then Sym(A) is a group with Id(A) as the indentity element
What is the Set Sn
The group Sym{1,2,…..,n}, the nth symmetric group
Theorem; Sn has order n!
Lemma; Disjoint Cycles Commute
What is a transposition. Write (a1,a2,….,am) as a product of transpositions
A cycle of length 2.
(a1,a2,….,am) = (a1,am)……(a1,a3)(a1,a2)
Define Pn, the nth alternating polynomial
Lemma; Let T be a transposition in Sn. Then T(Sn) = -Sn
Define the nth Alternating Subgroup An
Theorem; An is a subgroup of Sn
Theorem; Let n>1, then An has order equal to 1/2 Sn = n!/2
Define a Ring
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
Define a Subring and its 4 conditions
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
What is the easiest way to prove a set is a Ring
Prove it is a subring of a well known ring
Define a unit of a ring
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
What is a field?
A Ring such that every non-zero element is a unit
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
When is Z/mZ a field
m is prime
What are the gaussian integers Z[i]
a + bi
a,b in Z
