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1

binary operation

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

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2

associative

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

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3

commutative

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

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4

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)

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5

abelian group

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

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6

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

Check if ✩ is a binary operation.

Check if ✩ is associative.

Check identity (both ways).

Check inverse (both ways).

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7

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

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8

Prop 1: Identity and Inverse Properties

If (G, ✩) is a group

The identity of G is unique

For each a G, the inverse a^-1 is unique

(a^-1)^-1 = a for a in G

(ab)^-1 = b^-1a^-1

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