Chapter 1

Find the orders of the indicated elements in the indicated groups: (a) \(6 \in \mathbb{Z}_{10}\) (b) \(6 \in \mathbb{Z}_{15}\) (c) \(10 \in \mathbb{Z}_{42}\) (d) \(77 \in \mathbb{Z}_{210}\) (e) \(40 \in \mathbb{Z}_{210}\) (f) \(70 \in \mathbb{Z}_{210}\)

In Exercises 1 through 5 determine which of the indicated functions is a permutation of the indicated set. $$ f: \mathbb{R} \rightarrow \mathbb{R}, \text { where } f(x)=3 x+\sqrt{2} $$

In Exercises 1 through 10 find the order of the indicated element in the indicated group. $$ 2 \in \mathbb{Z}_{3} $$

Show that the indicated set \(G\) with the specified operation forms a group by showing that the four axioms in the definition of a group are satisfied. \(G=2 Z\) under addition

In Exercises 1 through 5 determine which of the indicated functions is a permutation of the indicated set. $$ f: \mathbb{R} \rightarrow \mathbb{R}, \text { where } f(x)=3 x^{2}+2 $$

Find the order of the indicated element in the indicated group. $$ 4 \in \mathbb{Z}_{10} $$

Let \(G\) be a group and \(a \in G\) an element of order \(|a|=6\). (a) Write all the elements of \(\langle a\rangle\). (b) Find in \(\langle a\rangle\) the elements \(a^{32}, a^{47}, a^{70}\).

Find all the generators of \(\mathbb{Z}_{10}, \mathbb{Z}_{12},\) and \(\mathbb{Z}_{15}\).

In Exercises 1 through 5 determine which of the indicated functions is a permutation of the indicated set. $$ f: U(5) \rightarrow U(5), \text { where } f(x)=x^{-1} $$

Show that the indicated set \(G\) with the specified operation forms a group by showing that the four axioms in the definition of a group are satisfied. \(G=C^{*}=C-\\{0\\}\) under complex multiplication

Let \(G=\langle a\rangle\) be a cyclic group of order \(30 .\) Find all the generators of \(G\).

In Exercises 1 through 5 determine which of the indicated functions is a permutation of the indicated set. $$ f: \mathbb{Z}_{6} \rightarrow \mathbb{Z}_{6}, \text { where } f(x)=x+3 $$

Show that the indicated set \(G\) with the specified operation forms a group by showing that the four axioms in the definition of a group are satisfied. \(G=G L(2, Q)\) under matrix multiplication

Draw the subgroup lattice diagram for \(\mathbb{Z}_{18}\).

In Exercises 6 through 9 find all the orbits of the indicated permutation. $$ \phi=\left(\begin{array}{lllllll} 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 7 & 6 & 3 & 2 & 1 & 4 & 5 \end{array}\right) $$

Find all the elements \(b \in \mathbb{Z}_{15}\) of order \(|b|=5\).

In Exercises 6 through 9 find all the orbits of the indicated permutation. $$ \phi=\left(\begin{array}{lllllllll} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ 6 & 5 & 9 & 4 & 1 & 8 & 7 & 2 & 3 \end{array}\right) $$

Find the order of the indicated element in the indicated group. $$ \mathbf{j} \in Q_{8} $$

Construct the group table for the indicated group, and determine whether or not it is Abelian. $$ G=D_{4} $$

Let \(G=\langle a\rangle\) be a cyclic group of order \(20 .\) Find all the elements \(b \in G\) of order \(|b|=10\).

Find the order of the indicated element in the indicated group. $$ -i \in \mathbb{C}^{*} $$

List all the cyclic subgroups of \(S_{3}\). Does \(S_{3}\) have a noncyclic proper subgroup?

In Exercises 6 through 9 find all the orbits of the indicated permutation. $$ \tau: \mathbb{Z} \rightarrow \mathbb{Z}, \text { where } \tau(x)=x-3 $$

Find the order of the indicated element in the indicated group. $$ -1+\sqrt{3} i \in \mathbb{C}^{*} $$

Construct the group table for the indicated group, and determine whether or not it is Abelian. $$ G=Q_{8} $$

List all the cyclic subgroups of \(D_{4}\). Does \(D_{4}\) have a noncyclic proper subgroup?

Let $$ \phi=\left(\begin{array}{llllllll} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ 3 & 8 & 4 & 1 & 6 & 7 & 2 & 5 \end{array}\right), \quad \tau=\left(\begin{array}{llllllll} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ 6 & 4 & 1 & 2 & 7 & 8 & 5 & 3 \end{array}\right) $$ Culculate: (a) \(\phi \tau\) and \(\tau \phi\) (b) \(\phi^{2} \tau\) and \(\phi \tau^{2}\) (c) the inverses \(\phi^{-1}\) and \(\tau^{-1}\) (d) the orders \(|\phi|\) and \(|\tau|\)

Find the order of the indicated element in the indicated group. $$ \cos (2 \pi / 7)+i \sin (2 \pi / 7) \in \mathrm{C}^{*} $$

Show that \(G L(2, Q)\) is non-Abelian.

In Exercises 11 through 13 express each permutation as a product of disjoint cycles, and then calculate its order. $$ \phi=\left(\begin{array}{llllllllll} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ 1 & 8 & 2 & 9 & 7 & 5 & 4 & 3 & 10 & 6 \end{array}\right) $$

In Exercises 11 through 19 give at least two examples of a nontrivial proper subgroup of the indicated group. $$ \mathbb{Z} $$

Show that if \(G\) is an Abelian group, then for all \(a, b \in G\) and for all integers \(n\), \((a b)^{n}=a^{n} b^{n}\)

Give examples of finite cyclic subgroups of \(\mathbb{C}^{*}\).

In Exercises 11 through 13 express each permutation as a product of disjoint cycles, and then calculate its order. $$ \phi=\left(\begin{array}{lllllll} 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 5 & 6 & 2 & 4 & 3 & 7 & 1 \end{array}\right)\left(\begin{array}{lllllll} 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 7 & 6 & 4 & 5 & 1 & 3 & 2 \end{array}\right) $$

In \(S_{3}\) find two elements \(a, b\) such that \((a b)^{2} \neq a^{2} b^{2}\).

In Exercises 11 through 13 express each permutation as a product of disjoint cycles, and then calculate its order. $$ \phi=\left(\begin{array}{lllll} 1 & 2 & 3 & 4 & 5 \\ 3 & 1 & 5 & 2 & 4 \end{array}\right)\left(\begin{array}{lllll} 1 & 2 & 3 & 4 & 5 \\ 2 & 4 & 1 & 5 & 3 \end{array}\right)\left(\begin{array}{lllll} 1 & 2 & 3 & 4 & 5 \\ 4 & 5 & 3 & 1 & 2 \end{array}\right) $$

In \(S_{3}\) find all elements a such that \(a^{2}=\rho_{0}=\) the identity, and all elements \(b\) such that \(b^{3}=\rho_{0}\).

Show that an \(n\) -cycle has order \(n\).

Find the inverse of each element of \(U(10)\) and of \(U(15)\).

Show that if \(\rho\) and \(\sigma\) in \(S_{n}\) are disjoint cycles, and \(\phi=\rho \sigma\), then \(|\phi|=\operatorname{lcm}(|\rho|,|\sigma|\) ).

Give at least two examples of a nontrivial proper subgroup of the indicated group. $$ S_{3} $$

Let \(G\) be the multiplicative group of all \(n\) th roots of unity. If \(a \in G,\) what is \(a^{1} ?\)

Show that every cyclic group is Abelian.

Show that an \(m\) -cycle is an even permutation if and only if \(m\) is odd.

Give at least two examples of a nontrivial proper subgroup of the indicated group. $$ D_{4} $$

Give an example of a group with the indicated combination of properties: (a) an infinite cyclic group (b) an infinite Abelian group that is not cyclic (c) a finite cyclic group with exactly six generators (d) a finite Abelian group that is not cyclic

Give at least two examples of a nontrivial proper subgroup of the indicated group. $$ 8 \mathbb{Z} $$

In the Klein 4 group, show that every element is equal to its own inverse.

Let \(H\) and \(K\) be cyclic subgroups of an Abelian group \(G,\) with \(|H|=10\) and \(|K|=14\). Show that \(G\) contains a cyclic subgroup of order 70 .

Explain why the set of odd permutations in \(S_{n}\) is not a subgroup of \(S_{n}\)