Abstract Algebra: Theory and Applications · Problem 4.5.1

Cyclic Groups — Thomas Judson

Mathematics — Abstract Algebra · cyclic-groups-prove-or-disprove

Problem

Prove or disprove each of the following statements. ((a)) All of the generators of

\mathbb{Z}_{60}

are prime. ((b))

U(8)

is cyclic. ((c))

\mathbb{Q}

is cyclic. ((d)) If every proper subgroup of a group

G

is cyclic, then

G

is a cyclic group. ((e)) A group with a finite number of subgroups is finite. Hint. (a) False; (c) false; (e) true.