Chapter 2: Groups  Definition and Examples of Groups  Elementary Properties of Groups.

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Transcript of Chapter 2: Groups  Definition and Examples of Groups  Elementary Properties of Groups.

Elementary Linear Algebra

Definition : Group

Abelian GroupA group G is called an Abelian Group ifab=ba for all elements a,b in G. G is called non Abelian Group if ab ba for some a,b in G.

Examples 1/

Examples 2/

Examples 2/

Multiplication table for {1,-1,i,-i}-ii-11-ii-111i-i1-1-11-1-iii-11i-i-iExamples 3/

Examples

examples

examplesThis is a non Abelian group

examples

examplesThe group U(n).

Note that U(p)={1,2,3,,p-1} if p is primeThe following examples are not groups:

examples

The group SL(2,F)Then SL(2,F) is a group under multiplication of matrices called the special linear group.For example SL(2,Z5)

The group GL(2, Z5)

In a group G, there is only one identityelement.