Class Definitions and Theorems

binary operation

A binary operation (✩) on a set G is a function G x G -> G

associative

We call ✩ associative if a✩(b✩c) = (a✩b)✩c for all a, b, c in G

commutative

We call ✩ commutative if a✩b= b✩a for all a,b in G

group

A group is an ordered pair (G, ✩) for a set G and a binary operation ✩: G x G -> G such that:
(i) ✩ is associative
(ii) there exists e in G called the identity, so that e✩g=g (left identity) and g✩e=g (right
identity) for all g in G.
(iii) for each g , there is an inverse g^-1in G so that (left inverse) g^-1✩g= e =g✩g^-1 (right inverse)

abelian group

A group (G, ✩) is called abelian if ✩ is commutative

(Steps) Determine if (G, ✩) is a group

1. Check if ✩ is a binary operation.
2. Check if ✩ is associative.
3. Check identity (both ways).
4. Check inverse (both ways).

Theorem 1: Cancellation Law

If (G, ✩) is a group and ab=cb or ba=bc, then a=c for all a,b,c in G

Prop 1: Identity and Inverse Properties

If (G, ✩) is a group
1. The identity of G is unique
2. For each a G, the inverse a^-1 is unique
3. (a^-1)^-1 = a for a in G
4. (ab)^-1 = b^-1a^-1