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
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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
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