Chapter 11

List the elements of \(Z_{2} \times Z_{4}\). Find the order of each of the elements. Is this group cyclic?

Find the order of the given element of the direct product. $$ (2,6) \text { in } Z_{4} \times Z_{12} $$

In Exercises 3 through 7, find the order of the given element of the direct product. 3\. \((2,6)\) in \(Z_{4} \times Z_{12}\)

Find the order of the given element of the direct product. $$ (2,3) \text { in } Z_{5} \times Z_{15} $$

Find the order of the given element of the direct product. $$ (3,10,9) \text { in } Z_{4} \times Z_{12} \times Z_{15} $$

Find the order of the given element of the direct product. $$ (3,6,12,16) \text { in } \mathrm{Z}_{4} \times \mathrm{Z}_{12} \times \mathrm{Z}_{2 \mathrm{\alpha}} \times \mathbb{Z}_{24} $$

What is the largest order among the orders of all the cyclic subgroups of \(Z_{6} \times Z_{s} ?\) of \(Z_{12} \times Z_{15} ?\)

Find all proper nontrivial subgroups of \(\mathrm{Z}_{2} \times \mathbb{Z}_{2}\).

Find all proper nontrivial subgroups of \(\mathrm{Z}_{2} \times \mathrm{Z}_{2} \times \mathrm{Z}_{2}\).

Find all subgroups of \(Z_{2} \times Z_{4}\) of order \(4 .\)

Find all subgroups of \(\mathrm{Z}_{2} \times \mathrm{Z}_{4}\) of order 4 .

Find all subgroups of \(Z_{2} \times \mathbb{Z}_{2} \times Z_{4}\) that are isomorphic to the Klein 4 . group.

Disregarding the order of the factors, write direct products of two or more groups of the form \(Z_{n}\) so that the, resulting product is isomorphic to \(Z_{\text {so }}\) in as many ways as possible.

Fill in the blanks. a. The cyclic subgroup of \(Z_{24}\) generated by 18 has order -. b. \(\mathrm{Z}_{3} \times \mathbb{Z}_{4}\) is of order - c. The element \((4,2)\) of \(\mathrm{Z}_{12} \times \mathrm{Z}_{4}\) has order.- d. The Klein 4 -group is isomorphic to \(Z\) Z \(\times Z\).. e. \(\mathrm{Z}_{2} \times \mathrm{Z} \times \mathrm{Z}_{4}\) has clements of finite order.

Find the maximum possible order for some element of \(\mathrm{Z}_{4} \times \mathbb{Z}_{6 .}\).

Are the groups \(\mathrm{Z}_{2} \times \mathrm{Z}_{12}\) and \(\mathrm{Z}_{4} \times \mathrm{Z}_{6}\) isomorphic? Why or why not?

Are the groups \(\mathrm{Z}_{2} \times \mathrm{Z}_{1 z}\) and \(\mathrm{Z}_{4} \times \mathrm{Z}_{6}\) isomorphic? Why or why not?

Find the maximum possible order for some element of \(Z_{8} \times Z_{10} \times Z_{24}\).

Are the groups \(\mathrm{Z}_{8} \times \mathrm{Z}_{10} \times \mathbb{Z}_{24}\) and \(\mathrm{Z}_{4} \times \mathrm{Z}_{12} \times \mathrm{Z}_{40}\) isomorphic? Why or why not?

Find the maximum possible order for some element of \(Z_{4} \times Z_{14} \times \mathbb{Z}_{15}\).

Are the groups \(\mathrm{Z}_{4} \times \mathrm{Z}_{1 \mathrm{~s}} \times \mathrm{Z}_{1 \mathrm{~s}}\) and \(\mathrm{Z}_{3} \times \mathrm{Z}_{36} \times \mathrm{Z}_{10}\) somorphic? Why or why not?

How many abelian groups (up to isomorphism) are there of order 24 ? of order 25 ? of order \((24)(25) ?\)

a. Let \(p\) be a prime number. Fill in the second row of the table to give the number of abelian groups of order \(p^{\prime \prime}\), up to isomorphism. \begin{tabular}{r|c|c|c|c|c|c|c|} \(n\) & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ \hline number of groups & & & & & & & \end{tabular} b. Let \(p, q\), and \(r\) be distinct prime numbers. Use the table you created to find the number of abelian groups, up to isomorphism, of the given order. i. \(p^{3} q^{4} r^{7}\) ii. \((q r)^{7}\) iii. \(q^{3} r^{4} q^{3}\)

Indicate schematically a Cayley digraph for \(Z_{9 n} \times Z_{n}\) for the generating set \(S=\\{(1,0),(0,1)\\}\).

Mark each of the following true or false. a. If \(G_{1}\) and \(G_{2}\) are any groups, then \(G_{1} \times G_{2}\) is always isomorphic to \(G_{2} \times G_{1}\). b. Computation in an external direct product of groups is casy if you know how to compute in each component group. c. Groups of finite onder must be used to form an external direct product. d. A group of prime onder could not be the internal direct product of two proper nontrivial subgroups, e. \(Z_{2} \times Z_{4}\) is isomorphic to \(Z_{8}\). f. \(\mathrm{Z}_{2} \times \mathbb{Z}_{4}\) is isomorphic to \(S_{\text {s. }}\). g. \(Z_{3} \times Z_{1}\) is isomorphic to \(S_{4}\). h. Every element in \(Z_{4} \times Z_{8}\) has order 8 . i. The order of \(Z_{12} \times \mathbb{Z}_{15}\) is 60 . j. \(Z_{m} \times Z_{n}\) has \(m n\) clements whether \(m\) and \(n\) are relatively prime or not.

Give an example illustrating that not every nontrivial abelian group is the internal direct product of two proper nontrivial subgroups.

a. How many subgroups of \(Z_{5} \times Z_{6}\) are isomorphic to \(Z_{3} \times Z_{6}\) ? b. How many subgroups of \(Z \times Z\) are isomorphic to \(Z \times \mathbb{Z} ?\)

Give an example of a nontrivial group that is not of prime order and is not the internal direct product of two nontrivial subgroups.

Mark each of the following true or false. a. Every abelian group of prime order is cyclic. b. Every abelian group of prime power order is cyclic. c. \(\mathrm{Z}_{\mathrm{s}}\) is generated by \(\\{4,6\\}\). d. \(Z_{1}\) is generated by \(\\{4,5,6\\}\). e. All finite abclian groups are classified up to isomorphism by Theorem 11.12. P. Any two finitely generated abelian groups with the same Betti number are isomorphic. g. Every abelian group of order divisible by 5 contains a cyclic subgroup of order 5 . h. Every abelian group of order divisible by 4 contains a cyclic subgroup of order \(4 .\) i. Every abelian group of order divisible by 6 contains a cyclic subgroup of order 6 . j. Every finite abelian group has a Betti number of \(0 .\)

Let \(G\) be an abelian group. Show that the elements of finite order in \(G\) form a subgroup. This subgroup is called the torsion subgroup of \(G\).

Find the order of the torsion subgroup of \(\mathrm{Z}_{4} \times \mathbb{Z} \times \mathrm{Z}_{1}\); of \(\mathrm{Z}_{12} \times \mathrm{Z} \times \mathrm{Z}_{12}\).

Find the torsion subgroup of the multiplicative group \(\mathrm{R}^{*}\) of nonrero real numbers.

Prove that a direct product of abelian groups is abelian.

Let \(G\) be an abelian group. Let \(H\) be the subset of \(G\) consisting of the identity \(e\) together with all elements of \(G\) of order 2 . Show that \(H\) is a subgroup of \(G\).

Let \(H\) and \(K\) be subgroups of a group \(G\). Exercises 50 and 51 ask you to establish necessary and sufficient criteria for \(G\) to appear as the internal direct product of \(H\) and \(K\). Let \(H\) and \(K\) be groups and let \(G=H \times K\). Recall that both \(H\) and \(K\) appear as subgroups of \(G\) in a natural way. Show that these subgroups \(H\) (actually \(H \times(e))\) and \(K\) (actually \(\\{e\\} \times K\) ) have the following properties. a. Every element of \(G\) is of the form \(h k\) for some \(h \in H\) and \(k \in K\). b. \(h k=k h\) for all \(h \in H\) and \(k \in K\). c. \(H \cap K=\\{e\\}\).

Show that a finite abelian group is not cyclic if and only if it contains a subgroup isomorphic to \(Z_{p} \times Z_{q}\) for some prime \(p\).

Prove that if a finite abelian group has order a power of a prime \(p\), then the order of every element in the group, is a power of \(p\). Can the hypothesis of commutativity be dropped? Why, or why not?

Let \(G, H\), and \(K\) be finitely generated abelian groups. Show that if \(G \times K\) is isomorphic to \(H \times K\), then \(G \simeq H\)