Abstract Algebra - Group Theory 1
Propositios for groups (2)
1. If au = av, then u = v 2. If ua = va, then u = v
Propositions for groups (1)
1. The identity element of groups is unique 2. The inverse for each element is unique 3. (a-1)-1 = a 4. (a⋆b)-1 = b-1 ⋆ a-1 5. for all a1, a2, a3,...,an ∈ G the product a1⋆a2⋆a3⋆...⋆an is not determined by parentheses (generalized associativity)
Associativity
A binary operation is associative if for all a,b,c ∈ G we have a ⋆ (b ⋆ c) = (a ⋆ b) ⋆ c
Binary operation
A function ⋆ such that ⋆: GxG → G. For a,b ∈ G we write a ⋆ b for ⋆(a,b)
group actions
A group action of group G on a set A is a map from G x A to A written as g*a 1) g1*(g2*a) = (g1 g2)*a 2) 1 * a = a
Groups
A group is an ordered pair (G, ⋆) where G is a set and ⋆ is a binary operation on G satisfying the following axioms: 1. (a ⋆ b) ⋆ c = a ⋆ (b ⋆ c) for all a,b,c ∈ G 2. There exists an identity element e ∈ G such that e ⋆ a = a ⋆ e = a for all a ∈ G 3. for each a ∈ G there exists some a-1 ∈ G such that a ⋆ a-1 = a-1 ⋆ a = e
Abelian Group
A group which also has commutativity a ⋆ b = b ⋆ a for all a,b ∈ G
Finite group
A group with finite elements
Isomorphism
A map from f:G to H is an isomorphism if f is a homomorphism and a bijection
Homomorphism
A map from group f:G to H such that f(xy) = f(x)f(y) for all x,y in G
Non isomorphic groups
Isomorphism requires 1) |G| = |H| 2) G is abelian iff H is abelian 3) for all x in G, |x| = |f(x)| Proving one of these false shows that groups are non-isomorphic
Order
The order of some element x ∈ G is defined as the smallest positive integer such that x^n = 1. This integer is denoted as |x|. If no such integer exists, then x is said to be of infinite order.
Symmetric Group
The set of permutations of sets
Dihedral group
The symmetries of an n-him denoted Dn with two fundamental types of elements, r: rotation clockwise by 2pi/n radians and s: reflection which when combined make up 2n different elements making Dn have order 2n
Commutativity
Two elements a,b ∈ G commute if a ⋆ b = b ⋆ a. If for all a,b ∈ G a ⋆ b = b ⋆ a then the binary operation is commutative
