Groups

One of the most fundamental algebraic structures is a group. Generally, groups are used to study the symmetries of objects and the solutions to equations.

A group G is a set with a binary operation * that satisfies the following properties:

  1. Closure: For all a, b in G, a * b is also in G.
  2. Associativity: For all a, b, c in G, (a * b) * c = a * (b * c).
  3. Identity Element: There exists an element e in G such that for all a in G, a * e = e * a = a.
  4. Inverse Element: For all a in G, there exists an element a' in G such that a * a' = a' * a = e.

Group Examples

  1. (Z, +) is a group where Z is the set of integers and + is the addition operation.
  • The identity element is 0.
  • Every element a has an inverse -a.
  • The operation is associative.
  • The set is closed under addition.
  1. (Q, +) is a group where Q is the set of rational numbers and + is the addition operation.

Theorems

Theorem: The identity element of a group is unique. Proof: Suppose there are two identity elements e and e'. Then, e = e * e' = e'.

Theorem: The inverse element of a group is unique. Proof: Suppose there are two inverse elements a' and a'' for an element a. Then, a' = a' * e = a' * (a * a'') = (a' * a) * a'' = e * a'' = a''.

Theorem (Cancelation Law): If a * b = a * c, then b = c. Proof: Multiply both sides by a' on the right. a * b = a * c a * b * a' = a * c * a' b * a' = c * a' b = c

Subgroups

Definition (Subgroup): A group (H, *) is a subgroup of a group (G, *) if H is a subset of G and H is a group under the same operation *.

TODO: Trivial subgroup TODO: Proper subgroup TODO: Cyclic subgroup TODO: Order of a subgroup TODO: Lagrange's theorem,

Cosets

Normal Subgroups

Quotient Groups

Group Actions

Homomorphisms and Isomorphisms

Direct Products