Homework SourseList all cyclic subgroups of U24SolutionThe group consists o
List all cyclic subgroups of U_24.![List all cyclic subgroups of U_24.SolutionThe group consists of the elements [i] belongsto [ Z/24Z] such that (i , 24) = 1. Thus the group is of order 8, consi](/WebImages/7/list-all-cyclic-subgroups-of-u24solutionthe-group-consists-o-989940-1761508931-0.webp "List all cyclic subgroups of U_24.SolutionThe group consists of the elements [i] belongsto [ Z/24Z] such that (i , 24) = 1. Thus the group is of order 8, consi")
Solution
The group consists of the elements [i] belongsto [ Z/24Z] such that (i , 24) = 1. Thus the group is of order 8, consisting of the elements [1]; [5]; [7]; [11]; [13]; [17]; [19]; [23].
The exponent of a group G is the smallest positive number n such that gn = e for
all g 2 G. For each of the 8 elements [i] 2 U24, one checks by direct computation thati2 1 mod 24. This shows the exponent is 2.
it is abelian, and of exponent 2, so has no element of
order 8 (= jU24j), hence is not cyclic.
