Let G be the subgroup of Sg generated by the permutations a= (13) (27), B= (1 7)(25)(39). ។ Throughout, you may use resu

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Let G Be The Subgroup Of Sg Generated By The Permutations A 13 27 B 1 7 25 39 Throughout You May Use Resu 1 (123.2 KiB) Viewed 29 times

Let G be the subgroup of Sg generated by the permutations a= (13) (27), B= (1 7)(25)(39). ។ Throughout, you may use results from lectures, provided that they are clearly stated. (a) What are the orders of aß and Ba? (b) Let H be the cyclic subgroup of G generated by the permutation (aB). Determine |H|, and show that H is not a normal subgroup of Sg. (c) By considering the number of elements of G calculated so far, as well as the orders of these elements, prove that |G| > 12. (You do not have to find all of the elements of G). - (d) Show that for any h E H, the conjugates a-1ha and ß-lhß are also elements of H. Briefly explain why this is sufficient to show that H is a normal subgroup of G. For the rest of this question, you are given that |G| = 12. =