Chapter 14

Computations In Exercises I through 8, find the order of the given factor group. 1\. \(\mathrm{Z}_{6} /(3)\)

In Exercises 1 through 8 , find the order of the given factor group. \(Z_{6} /\langle 3\rangle\)

In Exercises I through 8 , find the order of the given factor group. \(\left(\mathrm{Z}_{4} \times \mathbb{Z}_{12}\right) /((2\rangle \times\langle 2\rangle)\)

In Exercises I through 8 , find the order of the given factor group. \(\left(\mathrm{Z}_{4} \times \mathbb{Z}_{2}\right) /((2,1))\)

\(\left(Z_{3} \times Z_{5}\right) /\left((0) \times Z_{5}\right)\)

In Exercises I through 8 , find the order of the given factor group. \(\left(Z_{2} \times \mathbb{Z}_{4}\right) /((1,1)\\}\)

In Exercises 9 through 15, give the order of the element in the factor group. 9\. \(5+(4)\) in \(2_{12} /(4)\)

In Exercises 9 through 15, give the order of the element in the factor group. \((2,1)+((1,1))\) in \(\left(Z_{3} \times Z_{6}\right) /\langle(1,1)\rangle\)

\((2,0)+((4,4))\) in \(\left(Z_{6} \times Z_{8}\right) /((4,4))\)

In Exercises 9 through 15, give the order of the element in the factor group. \((2,0)+((4,4))\) in \(\left(Z_{6} \times Z_{8}\right) /\langle(4,4)\rangle\)

In Exercises 17 through 19, correct the definition of the italicized term without reference to the text, if correction is needed, so that it is in a form acceptable for publication. 17\. A normal subgroup \(H\) of \(G\) is one satisfying \(h G=G h\) for all \(h \in H\).

In Exercises 17 through 19 , correct the definition of the italicized term without reference to the text, if correction is needed, so that it is in a form acceptable for publication. A normal subgroup \(H\) of \(G\) is one satisfying \(g^{-1} h g \in H\) for all \(h \in H\) and all \(g \in G\).

An automorphism of a group \(G\) is a homomorphism mapping \(G\) into \(G\).

What is the importance of a normal subgroup of a group \(G\) ?

A student is asked to show that if \(H\) is a normal subgroup of an abelian group \(G\), then \(G / H\) is abelian. The student's proof starts as follows: We must show that \(G / H\) is abelian. Let \(a\) and \(b\) be two elements of \(G / H\). a. Why does the instructor reading this proof expect to find nonsense from here on in the student's paper? b. What should the student have written? c. Complete the proof.

Show that \(A_{n}\) is a normal subgroup of \(S_{n}\) and compute \(S_{n} / A_{n}\); that is, find a known group to which \(S_{n} / A_{n}\) is isomorphic.

Prove that the torsion subgroup \(T\) of an abelian group \(G\) is a normal subgroup of \(G\), and that \(G / T\) is torsion free. (See Exercise 22.)

A subgroup \(H\) is conjugate to a subgroup \(K\) of a group \(G\) if there exists an inner automorphism \(i_{z}\) of \(G\) such that \(i_{g}[H]=K\). Show that conjugacy is an equivalence relation on the collection of subgroups of \(G\).

Let \(H\) be a normal subgroup of a group \(G\), and let \(m=(G: H)\). Show that \(a^{m} \in H\) for every \(a \in G\).

Show that an intersection of normal subgroups of a group \(G\) is again a normal subgroup of \(G\).

Show that if a finite group \(G\) has exactly one subgroup \(H\) of a given order, then \(H\) is a normal subgroup of \(G\).

Show that if \(H\) and \(N\) are subgroups of a group \(G\), and \(N\) is normal in \(G\), then \(H \cap N\) is normal in \(H\). Show by an example that \(H \cap N\) need not be normal in \(G_{*}\)

Let \(G\) be a group containing at least one subgroup of a fixed finite order \(s\). Show that the intersection of all subgroups of \(G\) of order \(s\) is a normal subgroup of G. \([\) Hint: Use the fact that if \(H\) has order \(x\), then so does \(x^{-1} H x\) for all \(\left.x \in G .\right]\)

a. Show that all automorphisms of a group \(G\) form a group under function composition. b. Show that the inner automorphisms of a group \(G\) form a normal subgroup of the group of all automocptrisms of \(G\) under function composition. [Waming: Be sure to show that the inner automorphisms do form a subgroup.]

Show that the set of all \(g \in G\) such that \(i_{z}: G \rightarrow G\) is the identity inner automorphism \(i_{e}\) is a normal subgroup of a group \(G\).

Let \(G\) and \(G^{\prime}\) be groups, and let \(H\) and \(H^{\prime}\) be normal subgroups of \(G\) and \(G^{\prime}\), respectively. Let \(\phi\) be a homomorphism of \(G\) into \(G^{\prime}\). Show that \(\phi\) induces a natural homomorphism \(\phi_{n}:(G / H) \rightarrow\left(G^{\prime} / H^{\prime}\right)\) if \(\phi[H] \subseteq\) \(H^{\prime}\), (This fact is used constantly in algebraic topology.)

Use the properties \(\operatorname{det}(A B)=\operatorname{det}(A) \cdot \operatorname{det}(B)\) and \(\operatorname{det}\left(I_{n}\right)=1\) for \(n \times n\) matrices to show the following: a. The \(n \times n\) matrices with determinant 1 form a normal subgroup of \(G L(n, R)\). b. The \(n \times n\) matrices with determinant \(\pm 1\) form a normal subgroup of \(G L(n, \mathbb{R})\).