91Ó°ÊÓ

B. Elementary Properties of Cyclic Groups If \(G\) is a group of order \(n, G\) is cyclic iff \(G\) has an element of order \(n\).

Let \(G\) be a group and let \(a, b \in G\). Prove the following: If \(a\) is a power of \(b\), say \(a=b^{k}\), then \(\langle a\rangle \subseteq\langle b\rangle\).

Let \(G\) and \(H\) be groups, with \(a \in G\) and \(b \in H .\) Prove the following: If \((a, b)\) is a generator of \(G \times H\), then \(a\) is a generator of \(G\) and \(b\) is a generator of \(H .\)

A. Examples of Cyclic Groups List the elements of \(\langle 6\rangle\) in \(\mathbb{Z}_{16}\).

A. Examples of Cyclic Groups List the elements of \(\langle f\rangle\) in \(S_{6}\), where $$ f=\left(\begin{array}{llllll} 1 & 2 & 3 & 4 & 5 & 6 \\ 6 & 1 & 3 & 2 & 5 & 4 \end{array}\right) $$

Let \(\langle a\rangle\) be a cyclic group of order \(n\). For any integer \(k\), we may ask: which elements in \(\langle a\rangle\) have a \(k\) th root? The exercises which follow will answer this auestion. If \(m\) is a multiple of \(\operatorname{gcd}(k, n)\), then \(a^{m}\) has a \(k\) th root in \(\langle a\rangle .\) [HINT: Compute \(a^{m}\), and show that \(a^{m}=\left(a^{c}\right)^{k}\) for some \(\left.a^{c} \in\langle a\rangle .\right]\)

C. Generators of Cyclic Groups \(\langle a\rangle\) has \(\phi(n)\) different generators. [Use (1).]

Let \(G\) and \(H\) be groups, with \(a \in G\) and \(b \in H .\) Prove the following: If \(G \times H\) is a cyclic group, then \(G\) and \(H\) are both cyclic.

B. Elementary Properties of Cyclic Groups Every cyclic group is abelian. (HINT: Show that any two powers of \(a\) commute.)

Let \(G\) be a group and let \(a, b \in G\). Prove the following: Suppose \(a\) is a power of \(b\), say \(a=b^{k} .\) Then \(b\) is equal to a power of \(a\) iff \(\langle a\rangle=\langle b\rangle .\)