Fsc Part 1 Mathematics (Complete Solution)

Q5
Show that the set is an Abelia group w.r.t ordinary multiplication

Show that the set {1,ω,ω2}, when ω3=1,l is an Abelia group w.r.t ordinary multiplication

Solution:

1 ω ω2
1 1 ω ω2
ω ω ω2 1
ω2 ω2 1 ω

Suppose G={1,ω,ω2}

(i) A table show that all the entries belong to G.

(ii) Associative law holds in G w.r.t multiplication.

e.g. 1x(ωxω2)=1x1=1

     (1x ω)xω2=ωxω2=1

(iii) Since 1x1=1, inverse of 1 is 1

ω2xω=ωxω23=1, inverse of ω2 is ω and vice verse.

(iv) As table is symmetric about principal diagonal therefore commutative law holds in G.

 

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