hw04 - Show directly(i.e without arguing geometrically that...

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Modern Algebra 1 Homework 2.2 Spring 2010 Due January 27 Exercise 5. Let G be a group and x,y G . Prove that if xy = yx - 1 then x n y = yx - n for all n Z . [ Note: Induction can be used to prove this for all n N . Don’t forget to deal with the negative integers as well.] Exercise 6. Write out the Cayley (multiplication) table for D 3 , expressing every element in the form f i r j . Exercise 7.
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Unformatted text preview: Show directly (i.e. without arguing geometrically) that every element of D n of the form fr j is its own inverse. Exercise 8. Show that if x ∈ D n is not a rotation then rx = xr-1 . Exercise 9. If n = 2 k and z = r k , show that z is the only nonidentity element of D n that commutes with every other element....
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