Chapter 11

Let \([G ; *]\) be a group and \(a\) be an element of \(G .\) Define \(f: G \rightarrow G\) by \(f(x)=a * x\) (a) Prove that \(f\) is a bijection. (b) On the basis of part a, describe a set of bijections on the set of integers.

Determine the greatest common divisors of the following pairs of integers without using any computational assistance. (a) \(2^{3} \cdot 3^{2} \cdot 5\) and \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\) (b) \(7 !\) and \(3 \cdot 5 \cdot 7 \cdot 9 \cdot 11 \cdot 13\) (c) \(19^{4}\) and \(19^{5}\) (d) 12112 and 0

State whether each pair of groups below is isomorphic. For each pair that is, give an isomorphism; for those that are not, give your reason. (a) \(\mathbb{Z} \times \mathbb{R}\) and \(\mathbb{R} \times \mathbb{Z}\) (b) \(Z_{2} \times Z\) and \(Z \times \mathbb{Z}\) (c) \(\mathbb{R}\) and \(\mathbb{Q} \times \mathbf{Q}\) (d) \(\mathcal{P}(\\{1,2\\})\) with symmetric difference and \(\mathbb{Z}_{2}^{2}\) (e) \(\mathrm{E}_{2}^{2}\) and \(\mathbb{Z}_{4}\) (f) \(\mathbb{R}^{4}\) and \(M_{2 \times 2}(\mathbb{R})\) with matrix addition (g) \(\mathbb{R}^{2}\) and \(\mathbb{R} \times \mathbb{R}^{+}\) (h) \(\mathbb{Z}_{2}\) and the \(2 \times 2\) rook matrices (i) \(Z_{6}\) and \(Z_{2} \times Z_{s}\)

Which of the following subsets of the real numbers is a subgroup of \([\mathbb{R} ;+] ?\) (a) the rational numbers (b) the positive real numbers (c) \(\\{k / 2 \mid k\) is an integer \(\\}\) (d) \(\left\\{2^{k} \mid k\right.\) is an integer \(\\}\) (e) \(\\{x \mid-100 \leq x \leq 100\\}\)

Write out the group table of \(\mathbb{Z}_{2} \times \mathbb{Z}_{3}\) and find the two proper subgroups of this group.

Determine the properties that the following operations have on the positive integers. (a) addition (b) multiplication (c) \(M\) defined by \(a M b=\) larger of \(a\) and \(b\) (d) \(m\) defined by \(a m b=\) smaller of \(a\) and \(b\) (e) @ defined by \(a @ b=a^{b}\)

Find all possible values of the following, assuming that \(m\) is a positive integer. (a) \(\operatorname{gcd}(m+1, m)\) (b) \(\operatorname{gcd}(m+2, m)\) (c) \(\operatorname{gcd}(m+4, m)\)

Discuss the connection between groups and monoids. Is every monoid a group? Is every group a monoid?

Describe in simpler terms the following subgroups of \(\mathbb{Z}\) : (a) \(5 \mathbb{Z} \cap 4 \mathbb{Z}\) (b) \(4 \mathbb{Z} \cap 6 \mathbb{Z}\) (be careful) (c) the only finite subgroup of \(\mathbb{Z}\)

Prove that the relation "is isomorphic to" on groups is transitive.

List the additive inverses of the following elements: (a) 4,6,9 in \(\mathbb{Z}_{10}\) (b) 16,25,40 in \(\mathbb{Z}_{50}\)

Prove that, \(\oplus\), defined by \(A \oplus B=(A \cup B)-(A \cap B)\) is an associative oneration on \(\mathcal{P}\)

(a) Suppose that you were to be given a group \([G ; *]\) and asked to solve the equation \(x * x=e\). Without knowing the group, can you anticipate how many solutions there will be? (b) Answer the same question as part a for the equation \(x * x=x\).

In the group \(\mathbb{Z}_{11},\) what are: (a) 3(4)\(?\) (b) 36(4)\(?\) (c) How could you efficiently compute \(m(4), m \in \mathbb{Z} ?\)

The two groups \(\left[\mathbb{Z}_{4} ;+4\right]\) and \(\left[U_{5} ; \mathrm{x}_{5}\right]\) are isomorphic. One isomorphism \(T: \mathrm{Z}_{4} \rightarrow U_{5}\) is partially defined by \(T(1)=3 .\) Determine the values of \(T(0), T(2),\) and \(T(3)\)

(a) List the cyclic subgroups of \(\mathbb{Z}_{6}\) and draw an ordering diagram for the relation "is a subset of" on these subgroups. (b) Do the same for \(\mathbb{Z}_{12}\). (c) Do the same for \(\mathbb{Z}_{8}\). (d) On the basis of your results in parts a, b, and c, what would you expect if you did the same with \(\mathbb{Z}_{24} ?\)

Which of the following sets are subgroups of \(\mathbb{Z} \times \mathbb{Z} ?\) Give a reason for any negative answers. (a) \\{0\\} (b) \(\\{(2 j, 2 k) \mid j, k \in \mathbb{Z}\\}\) (c) \(\\{(2 j+1,2 k) \mid j, k \in \mathbb{Z}\\}\) (d) \(\left\\{\left(n, n^{2}\right) \mid n \in \mathbb{Z}\right\\}\) (e) \(\\{(j, k) \mid j+k\) is even \(\\}\)

Define \(a * b\) by \(|a-b|\), the absolute value of \(a-b\). Which properties does \(*\) have on the set of natural numbers, \(\mathbb{N} ?\)

The concept of a cyclic subgroup is a special case of the concept that we will discuss here. Let \([G ; *]\) be a group and \(S\) a nonempty subset of \(G\). Define the set \(\langle S\rangle\) recursively by: \- If \(a \in S,\) then \(a \in\langle S\rangle\) \- If \(a, b \in\langle S\rangle,\) then \(a * b \in\langle S\rangle,\) and \- If \(a \in\langle S\rangle,\) then \(a^{-1} \in\langle S\rangle\) (a) By its definition, \(\langle S\rangle\) has all of the properties needed to be a subgroup of \(G\). The only thing that isn't obvious is that the identity of \(G\) is in \(\langle S\rangle .\) Prove that the identity of \(G\) is in \(\langle S\rangle\). (b) What is \langle\\{9,15\\}\rangle in \([\mathbb{Z} ;+] ?\) (c) Prove that if \(H \leq G\) and \(S \subseteq H,\) then \(\langle S\rangle \leq H .\) This proves that \(\langle S\rangle\) is contained in every subgroup of \(G\) that contains \(S\); that is, \(\langle S\rangle=\bigcap \cap_{S \subseteq H, H \leq G} H\) (d) Describe \langle\\{0.5,3\\}\rangle in \(\left[\mathbb{R}^{+} ; \cdot\right]\) and in \([\mathbb{R} ;+]\). (e) If \(j, k \in \mathbb{Z},\langle\\{j, k\\}\rangle\) is a cyclic subgroup of \(\mathbb{Z}\). In terms of \(j\) and \(k\), what is a generator of \(\langle\\{j, k\\}\rangle ?\)

A student is asked to solve the following equations under the requirement that all arithmetic should be done in \(\mathbb{Z}_{2}\). List all solutions. (a) \(x^{2}+1=0\). (b) \(x^{2}+x+1=0\).

Prove that all infinite cyclic groups are isomorphic to \(\mathbb{Z}\).

Prove that if \(H, K \leq G,\) and \(H \cup K=G,\) then \(H=G\) or \(K=G\). Hint. Use an indirect argument.

(a) Write out the operation table for \(\times_{8}\) on \(\\{1,3,5,7\\},\) and convince your self that this is a group. (b) Let \(\mathbb{U}_{n}\) be the elements of \(\mathbb{Z}_{n}\) that have inverses with respect to \(\times_{n}\). Convince yourself that \(\mathbb{U}_{n}\) is a group under \(\times_{n}\). (c) Prove that the elements of \(\mathbb{U}_{n}\) are those elements \(a \in \mathbb{Z}_{n}\) such that \(\operatorname{gcd}(n, a)=1\). You may use Theorem 11.4.9 in this proof.

Prove that if \(G\) is any group and \(g\) is some fixed element of \(G\), then the function \(\phi_{g}\) defined by \(\phi_{g}(x)=g * x * g^{-1}\) is an isomorphism from \(G\) into itself. An isomorphism of this type is called an inner automorphism.

It can be shown that there are five non-isomorphic groups of order eight. You should be able to describe at least three of them. Do so without use of tables. Be sure to explain why they are not isomorphic.

Suppose \(f: \mathbb{Z}_{17} \rightarrow \mathbb{Z}_{17}\) such \(f(i)=a \times_{17} i+{ }_{17} b\) where \(a\) and \(b\) are integer constants. Furthermore, assume that \(f(1)=11\) and \(f(2)=4\). Find a formula for \(f(i)\) and also find a formula for the inverse of \(f\).