91Ó°ÊÓ

In Exercises 1 through 4 , find the quotient and remainder, according to the division algorithm, when \(n\) is divided by \(m\). $$ n=42, m=9 $$

Find the quotient and remainder, according to the division algorithm, when \(n\) is divided by \(m\). $$ n=50, m=8 $$

In Exercises 8 through 11 , find the number of generators of a cyclic group having the given order. 5

find the number of genentors of a cyclic group having the given order. $$ 8 $$

Find the number of generators of a cyclic group having the given order. $$ 12 $$

Find the number of automorphisms of the given group. $$ Z_{12} $$

Find all orders of subgroups of the given group. $$ Z_{6} $$

Mark each of the following true or false. a. Every cyclic group is abelian. b. Every abellan group is cyclic. c. \(Q\) under addition is a cyclic group. d. Every element of every cyclic group generates the group. e. There is at least one abelian group of every finite order \(>0\). f. Every group of order \(\leq 4\) is cyclic. g. All generators of \(Z_{20}\) are prime numbers. h. If \(G\) and \(G^{\prime}\) are groups, then \(G \cap G^{\prime}\) is a group. i. If \(H\) and \(K\) are subgroups of a group \(G\), then \(H \cap K\) is a group. J. Every cyclic group of order \(>2\) has at least two distinct generators.

In Exercises 33 through 37 , either give an example of a group with the property described, or explain why no example exists. A finite group that is not cyclic

Either give an example of a group with the property described, or explain why no example exists. An infinite group that is not cyclic