ass3_1 - K . What is the order of K ? Is K abelian or...

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MATH 335 001 Computational Algebra Winter 2006 Assignment 3.1 GROUPS 1) Find a complete rewriting system for the group G = h x,y,z | x 3 = 1 ,y 2 = 1 ,xy = yx 2 ,z 2 = 1 ,xz = zx,zy = yz i . Show that | G | = 12. 2) Show that the group H = h x,y | x n = 1 ,y 2 = 1 ,xy = yx - 1 i has order 2 n and is isomorphic to the multiplicative group of 2 × 2 matrices of the form ± ε k 0 1 where ε ∈ { 1 , - 1 } ,k Z n . 3) Let K = h x,y | x 4 = 1 ,x 2 = y 2 ,xy = yx 3 i . Find a set of normal forms for the group
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Unformatted text preview: K . What is the order of K ? Is K abelian or nilpotent? 4) Show that the group L = h x,y | x 3 = 1 ,y 2 = 1 , ( xy ) 3 = 1 i . is isomorphic to the group A 4 , which is a subgroup of the symmetric group S 4 , consisting of all even permutations....
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