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# Groups, Order and Commutativity

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1. If x is an element of a group and x is of order n then the elements 1, x, x^2,...x^n-1 are distinct (don't know how to show this!)

2. Let Y=<u,v/u^4=v^3=1, uv=v^2u^2> Y here is a group show
a) v^2=v^-1
b) v commutes with u^3
c) v commutes with u
d)uv=1
e)u=1, deduce that v=1 and conclude that Y=1

https://brainmass.com/math/group-theory/groups-order-and-commutativity-94829

#### Solution Preview

1. Proof:
If 1,x,x^2,...,x^(n-1) is not distinct, then we can find some 0<=i<j<=n-1, such that x^i = x^j. Then ...

#### Solution Summary

Groups, Order and Commutativity are investigated.

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