Abstract Algebra: Theory and Applications · Problem 3

Permutation Groups — Thomas Judson

Mathematics — Abstract Algebra · cyclic-groups-subgroups

Problem

Find all cyclic subgroups of each of the following groups. ((g)) The subgroup generated by 3 in

U(20)

((h)) The subgroup generated by 5 in

U(18)

((i)) The subgroup of

\mathbb{R}^*

generated by 7 ((j)) The subgroup of

\mathbb{C}^*

generated by

i

where

i^2 = -1

((k)) The subgroup of

\mathbb{C}^*

generated by

2i

((l)) The subgroup of

\mathbb{C}^*

generated by

(1+i)/\sqrt{2}

((m)) The subgroup of

\mathbb{C}^*

generated by

(1+\sqrt{3}\,i)/2

Hint. (g)

\{1,3,7,9\}

; (j)

\{1,-1,i,-i\}