Chapter 11

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 \(H .\)

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 \(\langle a\rangle \subseteq\langle b\rangle\).

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

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

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 \(\left.a^{c} \in\langle a\rangle .\right]\)

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.

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 \(\langle a\rangle=\langle b\rangle .\)

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

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

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 \(G\) be a group and let \(a, b \in G\). Prove the following: Suppose \(a \in\langle b\rangle\). Then \(\langle a\rangle=\langle b\rangle\) iff \(a\) and \(b\) have the same order.

B. Elementary Properties of Cyclic Groups If \(G=\langle a\rangle\) and \(b \in G\), the order of \(b\) is a factor of the order of \(a\).

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. \(a\) has a \(k\) th root in \(\langle a\rangle\) iff \(k\) and \(n\) are relatively prime.

Let \(G\) be a group and let \(a, b \in G\). Prove the following: Let \(\operatorname{ord}(a)=n\), and \(b=a^{k}\). Then \(\langle a\rangle=\langle b\rangle\) iff \(n\) and \(k\) are relatively prime.

B. Elementary Properties of Cyclic Groups In any cyclic group of order \(n\), there are elements of order \(k\) for every integer \(k\) which divides \(n\).

A. Examples of Cyclic Groups If \(f(x)=x+1\), describe the cyclic subgroup \(\langle f\rangle\) of \(S_{8}\)

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. Let \(p\) be a prime number. (i) If \(n\) is not a multiple of \(p\), then every element in \(\langle a\rangle\) has a \(p\) th root. (ii) If \(n\) is a multiple of \(p\), and \(a^{m}\) has a \(p\) th root, then \(m\) is a multiple of \(p\). (Thus, the only elements in \(\langle a\rangle\) which have \(p\) th roots are \(e, a^{p}, a^{2 p}\), etc.)

Let \(G\) be a group and let \(a, b \in G\). Prove the following: Let \(\operatorname{ord}(a)=n\), and suppose \(a\) has a \(k\) th root, say \(a=b^{k} .\) Then \(\langle a\rangle=\langle b\rangle\) iff \(k\) and \(n\) are relatively prime.

C. Generators of Cyclic Groups An element \(x\) in \(\langle a\rangle\) has order \(m\) iff \(x\) is a generator of \(C_{m}\).

Let \(G\) and \(H\) be groups, with \(a \in G\) and \(b \in H .\) Prove the following: Suppose \((c, d) \in G \times H\), where \(c\) has order \(m\) and \(d\) has order \(n .\) If \(m\) and \(n\) are not relatively prime (hence have a common factor \(q>1)\), then the order of \((c, d)\) is less than \(m n\).

A. Examples of Cyclic Groups Show that \(-1\), as well as 1 , is a generator of \(\mathbb{Z} .\) Are there any other generators of \(\mathbb{Z} ?\) Explain! What are the generators of an arbitrary infinite cyclic group \(\langle a\rangle ?\)

A. Examples of Cyclic Groups Is \(\mathbb{R}^{*}\) cyclic? Try to prove your answer. $$ \text { (HINT: If } k<1, \text { then } k>k^{2}>k^{3}>\ldots ; \quad k^{3} \quad k^{2} \quad k \quad 1 $$ if \(k>1\), then \(k

Let \(G\) and \(H\) be groups, with \(a \in G\) and \(b \in H .\) Prove the following: Let \(\langle a\rangle\) be a cyclic group of order \(m n\), where \(m\) and \(n\) are relatively prime. Then \(\langle a\rangle \cong\left\langle a^{m}\right\rangle \times\left\langle a^{n}\right\rangle\)