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Permutation Groups and Commutativity

Let Y=(u v/u^4=v^3=1,uv=u^2v^2)

Show that
a) v^2=v^-1
b) v commutes with u^3
c)u commutes with u
d)uv=1
e)show that u=1, deduce that v=1 and conclude that Y=1

Solution Preview

Proof:
(a) Since v^3=1, then v^(-1)*v^3=v^(-1), thus v^2=v^(-1)
(b) Since uv=u^2v^2, then u^(-1)uvv^(-1)=u^(-1)u^2v^2v^(-1), then 1=uv. So we get uv=1. Then v=u^(-1). Thus ...

Solution Summary

Permutation Groups and Commutativity are investigated. The solution is detailed and well presented.

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