Consider the group A5. (a) Compute a = (1,2, 3)(3, 4, 5) and ß = (1, 2, 3)(2, 3, 4). (b) List possible types of cyclic decompositions for elements in A5. (c) Show that for any distinct a, b, c E {1,2,3, 4, 5} there is y E A5 so that y(1,2, 3)y-1 = (a, b, c). (d) Deduce that if a normal subgroup N 4 A5 contains a 3-cycle (a, b, c), then N = A5. %3|

Consider the group A5. (a) Compute a = (1,2, 3)(3, 4, 5) and ß = (1, 2, 3)(2, 3, 4). (b) List possible types of cyclic decompositions for elements in A5. (c) Show that for any distinct a, b, c E {1,2,3, 4, 5} there is y E A5 so that y(1,2, 3)y-1 = (a, b, c). (d) Deduce that if a normal subgroup N 4 A5 contains a 3-cycle (a, b, c), then N = A5. %3|

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter4: More On Groups

Section4.4: Cosets Of A Subgroup

Problem 17E: Show that a group of order 4 either is cyclic or is isomorphic to the Klein four group e,a,b,ab=ba.

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