Only d & e For each of the following, write TRUE if the statement is always true. Otherwise, write FALSE.(a) If a E An, then (a) < An-(b) The element 9 is a generator for Z57.(c) In a finite group G, there exists an element of order d if and only if d divides |G|.(d) A cyclic group of order n has no proper nontrivial subgroup if and only if n is prime.(e) If o e Sn such that o = T1T2 · · Tk where each T; for i = 1,2, .., k is a transposition, then for anypositive integer n, o" = Tf'T" . .- T.

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Answered: (d) A cyclic group of order n has no…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter3: Groups

Section3.1: Definition Of A Group

Problem 7TFE: Label each of the following statements as either true or false. The Cayley table for a group will...

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(d) A cyclic group of order n has no proper nontrivial subgroup if and only if n is prime. (e) If o e Sn such that o = T¡T2• · Tk where each T; for i = 1,2, ., k is a transposition, then for any positive integer n, o" = T{"T" - · T.