Chapter 10

Let \(a\) denote an element of a group \(G\). Let \(a\) have order 12 . Prove that if \(a\) has a cube root, say \(a=b^{3}\) for some \(b \in G\), then \(b\) has order 36. \(\left\\{\mathrm{HINT}\right.\) : Show that \(b^{36}=e\); then show that for each factor \(k\) of \(36, b^{k}=e\) is impossible. [Example: If \(b^{12}=e\), then \(\left.b^{12}=\left(b^{3}\right)^{4}=a^{4}=e .\right]\) Derive your conclusion from these facts. \(\\}\)

Let \(a\) and \(b\) be elements of a group \(G .\) Let ord \((a)=m\) and \(\operatorname{ord}(b)=n ; \operatorname{lcm}(m, n)\) denotes the least common multiple of \(m\) and \(n .\) Prove the following: If \(a\) and \(b\) commute, then ord \((a b)\) is a divisor of \(\operatorname{lcm}(m, n)\).

Let \(a, b\), and \(c\) be elements of a group \(G\). Prove the following: \(\operatorname{Ord}(a)=1 \quad\) iff \(\quad a=e\)

What is the order of 10 in \(\mathbb{Z}_{25}\) ?

Prove that \(a^{m} a^{n}=a^{m+n}\) in the following cases: (i) \(m=0\) (ii) \(m<0\) and \(n>0\) (iii) \(m<0\) and \(n<0\)

Let \(a\) denote an element of a group \(G\). Let \(a\) have order 6 . If \(a\) has a fourth root in \(G\), say \(a=b^{4}\), what is the order of \(b ?\)

From elementary arithmetic we know that every integer may be written uniquely as a product of prime numbers. Two integers \(m\) and \(n\) are said to be relatively prime if they have no prime factors in common. (For example, 15 and 8 are relatively prime.) Here is a useful fact: If \(m\) and \(n\) are relatively prime, and \(m\) is a factor of \(n k\), then \(m\) is a factor of \(k\). (Indeed, all the prime factors of \(m\) are factors of \(n k\) but not of \(n\), hence are factors of \(k\).) Let \(a\) be an element of order \(n\) in a group \(G\). Prove the following: If \(a^{m}\) has order \(n\), then \(m\) and \(n\) are relatively prime. [HINT: Assume \(m\) and \(n\) have a common factor \(q>1\), hence we can write \(m=m^{\prime} q\) and \(n=n^{\prime} q\). Explain why \(\left(a^{m}\right)^{n^{\prime}}=e\), and proceed from there.]

Let \(a\) be an element of order 12 in a group \(G\) What is the order of \(a^{8}\) ?

Let \(a\) and \(b\) be elements of a group \(G .\) Let ord \((a)=m\) and \(\operatorname{ord}(b)=n ; \operatorname{lcm}(m, n)\) denotes the least common multiple of \(m\) and \(n .\) Prove the following: 2 If \(m\) and \(n\) are relatively prime, then no power of \(a\) can be equal to any power of \(b\) (except for \(e\) ). (REMARK: Two integers are said to be relatively prime if they have no common factors except \(\pm 1 .\) )

Let \(a\) be any element of a group \(G\). Prove the following: The order of \(a^{k}\) is a divisor (factor) of the order of \(a\).

Let \(a, b\), and \(c\) be elements of a group \(G\). Prove the following: If \(\operatorname{ord}(a)=n\), then \(a^{n-r}=\left(a^{r}\right)^{-1}\)

What is the order of 6 in \(\mathbb{Z}_{16}\) ?

Let \(a\) denote an element of a group \(G\). Let \(a\) have order 10. If \(a\) has a sixth root in \(G\), say \(a=b^{6}\), what is the order of \(b\) ?

Let \(a\) be an element of order 12 in a group \(G\) What are the orders of \(a^{9}, a^{10}, a^{5}\) ?

Let \(a\) and \(b\) be elements of a group \(G .\) Let ord \((a)=m\) and \(\operatorname{ord}(b)=n ; \operatorname{lcm}(m, n)\) denotes the least common multiple of \(m\) and \(n .\) Prove the following: If \(m\) and \(n\) are relatively prime, then the products \(a^{i} b^{j}(0 \leqslant i \leqslant m, 0 \leqslant j \leqslant n)\) are all distinct.

Let \(a\) be any element of a group \(G\). Prove the following: If \(\operatorname{ord}(a)=k m\), then \(\operatorname{ord}\left(a^{k}\right)=m\).

Let \(a, b\), and \(c\) be elements of a group \(G\). Prove the following: If \(a^{k}=e\) where \(k\) is odd, then the order of \(a\) is odd.

What is the order of $$ f=\left(\begin{array}{llllll} 1 & 2 & 3 & 4 & 5 & 6 \\ 6 & 1 & 3 & 2 & 5 & 4 \end{array}\right) $$ in \(S_{6}\) ?

Let \(a\) denote an element of a group \(G\). Let \(a\) have order \(n\), and suppose \(a\) has a \(k\) th root in \(G\), say \(a=b^{k} .\) Explain why the order of \(b\) is a factor of \(n k\). Let $$ \operatorname{ord}(b)=\frac{n k}{l} $$

From elementary arithmetic we know that every integer may be written uniquely as a product of prime numbers. Two integers \(m\) and \(n\) are said to be relatively prime if they have no prime factors in common. (For example, 15 and 8 are relatively prime.) Here is a useful fact: If \(m\) and \(n\) are relatively prime, and \(m\) is a factor of \(n k\), then \(m\) is a factor of \(k\). (Indeed, all the prime factors of \(m\) are factors of \(n k\) but not of \(n\), hence are factors of \(k\).) Let \(a\) be an element of order \(n\) in a group \(G\). Prove the following: Let \(l\) be the least common multiple of \(m\) and \(n\). Let \(l / m=k\). Explain why \(\left(a^{m}\right)^{k}=e\).

Let \(a\) be an element of order 12 in a group \(G\) Which powers of \(a\) have the same order as \(a\) ? [That is, for what values of \(k\) is \(\left.\operatorname{ord}\left(a^{k}\right)=12 ?\right]\)

Let \(a\) and \(b\) be elements of a group \(G .\) Let ord \((a)=m\) and \(\operatorname{ord}(b)=n ; \operatorname{lcm}(m, n)\) denotes the least common multiple of \(m\) and \(n .\) Prove the following: 4 Let \(a\) and \(b\) commute. If \(m\) and \(n\) are relatively prime, then ord \((a b)=m n\).

Let \(a\) be any element of a group \(G\). Prove the following: If \(\operatorname{ord}(a)=n\) where \(n\) is odd, then ord \(\left(a^{2}\right)=n\)

Let \(a, b\), and \(c\) be elements of a group \(G\). Prove the following: \(\operatorname{Ord}(a)=\operatorname{ord}\left(b a b^{-1}\right)\)

What is the order of 1 in \(\mathbb{R}^{*} ?\) What is the order of 1 in \(\mathbb{R} ?\)

Let \(a\) and \(b\) be elements of a group \(G .\) Let ord \((a)=m\) and \(\operatorname{ord}(b)=n ; \operatorname{lcm}(m, n)\) denotes the least common multiple of \(m\) and \(n .\) Prove the following: Let \(a\) and \(b\) commute. There is an element \(c\) in \(G\) whose order is \(\operatorname{lcm}(m, n)\).

Let \(a\) be any element of a group \(G\). Prove the following: If \(a\) has order \(n\), and \(a^{r}=a^{s}\), then \(n\) is a factor of \(r-s\)

Let \(a, b\), and \(c\) be elements of a group \(G\). Prove the following: The order of \(a^{-1}\) is the same as the order of \(a\).

If \(A\) is the set of all the real numbers \(x \neq 0,1,2\), what is the order of $$ f(x)=\frac{2}{2-x} $$ in \(S_{A}\) ?

Let \(a\) denote an element of a group \(G\). Let \(a\) have order \(n .\) Let \(k\) be an integer such that every prime factor of \(k\) is a factor of \(n\). Prove: If \(a\) has a \(k\) th root \(b\), then ord \((b)=n k\).

In \(\mathbb{Z}_{24}\), list all the elements \((a)\) of order \(2 ;(b)\) of order \(3 ;(c)\) of order \(4 ;(d)\) of order 6