Prove that an Abelian group of order 33 is cyclic.

Problem 79E Let G be a group of order 100 that has exactly one subgroup of order 5. Prove that it has a subgroup of order 10.

Problem 5E Show that if G is the internal direct product of H1, H2, . . . , Hn and i ? j with 1 ? i ? n, 1 ? j ? n, then Hi ? Hj = {e}. (This exercise is referred to in this chapter.)

Problem 2E Prove that An is normal in Sn.

Problem 1E Let H = {(1), (12)}. Is H normal in S3?

Problem 3E In D4, let K = {R0, R90, R180, R270}. Write HR90 in the form xH, where x ? K. Write DR270 in the form xD, where x ? K. Write R90V in the form Vx, where x ? K.

Write (12)(13)(14) in the form \(\alpha(12)\), where \(\alpha \in A_4\). Write (1234) (12)(23), in the form \(\alpha(1234)\), where \(\alpha \in A_4\).

Problem 6E Let . Is H a normal subgroup of GL(2, R)?

Let \(G=G L(2, \mathbf{R})\) and let K be a subgroup of \(\mathbf{R}^*\). Prove that \(H=\{A \in G \mid \operatorname{det} A \in K\}\) is a normal subgroup of G.

Viewing \(\langle 3 \rangle\) and \(\langle 12 \rangle\) as subgroups of Z, prove that \(\langle 3 \rangle / \langle 12 \rangle\) is isomorphic to \(Z_4\). Similarly, prove that \(\langle 8 \rangle / \langle 48 \rangle\) is isomorphic to \(Z_6\). Generalize to arbitrary integers k and n.

Problem 9E Prove that if H has index 2 in G, then H is normal in G. (This exercise is referred to in Chapters 24 and 25 and this chapter.)

Problem 10E Let H = {(1), (12)(34)} in A4. a. Show that H is not normal in A4. b. Referring to the multiplication table for A4 in Table 5.1 on page 111, show that, although ?6H = ?7H and ?9H = ?11H, it is not true that ?6?9H = ?7?11H. Explain why this proves that the left cosets of H do not form a group under coset multiplication.

Prove that a factor group of a cyclic group is cyclic.

Problem 13E Prove that a factor group of an Abelian group is Abelian.

Problem 11E Let . Show that G/H is not isomorphic to G/K. (This shows that H ? K does not imply that G/H ? G/K.)

What is the order of the element \(14+ \langle 8 \rangle\) in the factor group \(Z_{24}/ \langle 8 \rangle\)?

Problem 15E What is the order of the element 4U5(105) in the factor group U(105)/U5(105)?

Problem 16E Recall that Z(D6) = {R0, R180}. What is the order of the element R60Z(D6) in the factor group D6/Z(D6)?

Let \(G=Z /\langle 20\rangle \text { and } H=\langle 4\rangle\langle\langle 20\rangle\). List the elements of H and G/H.

What is the order of the factor group \(Z_{60} / \langle 15 \rangle\)?

Problem 19E What is the order of the factor group

Problem 20E Construct the Cayley table for U(20)/U5(20).

Prove that an Abelian group of order 33 is cyclic.

Problem 22E Determine the order of . Is the group cyclic?

Problem 23E Determine the order of . Is the group cyclic?

Problem 24E The group is isomorphic to one of Determine which one by elimination.

Problem 25E Let G = U(32) and H = {1, 31}. The group G/H is isomorphic to one of . Determine which one by elimination.

Problem 26E Let G be the group of quaternions given by the table in Exercise 4 of the Supplementary Exercises for Chapters 1–4, and let H be the subgroup {e, a2}. Is G/H isomorphic to

Let G = U(16), H = {1, 15}, and K = {1, 9}. Are H and K isomorphic? Are G/H and G/K isomorphic?

Let \(G=Z_4 \oplus Z_4, H= \{(0, 0), (2, 0), (0, 2), (2, 2)\}\), and \(K= \langle (1,2) \rangle\). Is G/H isomorphic to \(Z_4\) or \(Z_2 \oplus Z_2\)? Is G/K isomorphic to \(Z_4\) or \(Z_2 \oplus Z_2\)?

Prove that \(A_{4} \oplus Z_{3}\) has no subgroup of order 18.

Problem 30E Express U(165) as an internal direct product of proper subgroups in four different ways.

Let R* denote the group of all nonzero real numbers under multiplication. Let \(\mathbf{R}^+\) denote the group of positive real numbers under multiplication. Prove that R* is the internal direct product of \(\mathbf{R}^+\) and the subgroup {1, 21}.

Problem 33E Let H and K be subgroups of a group G. If G = HK and g = hk, where h ? H and k ? K, is there any relationship among |g|, |h|, and |k|? What if G = H × K?

Prove that \(D_4\) cannot be expressed as an internal direct product of two proper subgroups.

Problem 34E In Z, let . Prove that Z = HK. Does Z = H × K?

Let \(G=\left\{3^{a} 6^{b} 10^{c} \mid a, b, c \in Z\right\}\) under multiplication and \(H=\left\{3^{a} 6^{b} 12^{c} \mid a, b, c \in Z\right\}\) under multiplication. Prove that \(G = \langle 3\rangle \times\langle 6\rangle \times\langle 10\rangle\), whereas \(H \neq\langle 3\rangle \times\langle 6\rangle \times\langle 12\rangle\).

Problem 36E Determine all subgroups of R* (nonzero reals under multiplication) of index 2.

Problem 37E Let G be a finite group and let H be a normal subgroup of G. Prove that the order of the element gH in G/H must divide the order of g in G.

Problem 39E If H is a normal subgroup of a group G, prove that C(H), the centralizer of H in G, is a normal subgroup of G.

Problem 38E Let H be a normal subgroup of G and let a belong to G. If the element aH has order 3 in the group G/H and |H| =10, what are the possibilities for the order of a?

Problem 40E Let ? be an isomorphism from a group G onto a group . Prove that if H is a normal subgroup of G, then ?(H) is a normal subgroup of G.

Problem 41E Show that Q, the group of rational numbers under addition, has no proper subgroup of finite index.

An element is called a square if it can be expressed in the form b2 for some b. Suppose that G is an Abelian group and H is a subgroup of G. If every element of H is a square and every element of G/H is a square, prove that every element of G is a square. Does your proof remain valid when “square” is replaced by “nth power,” where n is any integer?

Show, by example, that in a factor group G/H it can happen that \(aH=bH\) but \(|a| \ne |b|\).

Problem 44E Observe from the table for A4 given in Table 5.1 on page 111 that the subgroup given in Example 9 of this chapter is the only subgroup of A4 of order 4. Why does this imply that this subgroup must be normal in A4? Generalize this to arbitrary finite groups.

Problem 45E Let p be a prime. Show that if H is a subgroup of a group of order 2p that is not normal, then H has order 2.

Problem 47E Suppose that N is a normal subgroup of a finite group G and H is a subgroup of G. If |G/N | is prime, prove that H is contained in N or that NH = G.

Problem 48E If G is a group and |G: Z(G)| = 4, prove that .

Problem 49E Suppose that G is a non-Abelian group of order p3, where p is a prime, and Z(G) ? {e}. Prove that |Z(G)| = p.

Show that \(D_{13}\) is isomorphic to Inn\((D_{13})\).

If |G| = pq, where p and q are primes that are not necessarily distinct, prove that |Z(G)| = 1 or pq.

Problem 53E Determine all subgroups of R* that have finite index.

Let \(G=\{\pm 1, \pm i, \pm j, \pm k\}\), where \(i^2=j^2=k^2=-1,-i=(-1) i, 1^2=(-1)^2=1, i j=-j i=k, j k=-k j=i\), and ki = -ik=j. a. Construct the Cayley table for G. b. Show that \(H=\{1,-1\}<G\). c. Construct the Cayley table for G/H. Is G/H isomorphic to \(Z_4\) or \(Z_2 \oplus Z_2\)? (The rules involving i, j, and k can be remembered by using the circle below. Going clockwise, the product of two consecutive elements is the third one. The same is true for going counterclockwise, except that we obtain the negative of the third element.) This is the group of quaternions that was given in another form in Exercise 4 in the Supplementary Exercises for Chapters 1-4. It was invented by William Hamilton in 1843. The quaternions are used to describe rotations in three dimensional space, and they are used in physics. The quaternions can be used to extend the complex numbers in a natural way.

Problem 51E Let N be a normal subgroup of G and let H be a subgroup of G. If N is a subgroup of H, prove that H/N is a normal subgroup of G/N if and only if H is a normal subgroup of G.

Problem 56E Show that the intersection of two normal subgroups of G is a normal subgroup of G. Generalize.

Problem 52E Let G be an Abelian group and let H be the subgroup consisting of all elements of G that have finite order. (See Exercise 20 in the Supplementary Exercises for Chapters 1–4.) Prove that every nonidentity element in G/H has infinite order.

Problem 55E In D4, let K = {R0, D} and let L = {R0, D, D ', R180}. Show that but that K is not normal in D4. (Normality is not transitive. Compare Exercise 4, Supplementary Exercises for Chapters 5–8.)

Give an example of subgroups H and K of a group G such that HK is not a subgroup of G.

Problem 59E Let N be a normal subgroup of a group G. If N is cyclic, prove that every subgroup of N is also normal in G. (This exercise is referred to in Chapter 24.)

Problem 58E If N and M are normal subgroups of G, prove that NM is also a normal subgroup of G.

Let H be a normal subgroup of a finite group G and let \(x \in G\). If \(\operatorname{gcd}(|x|,|G / H|)=1\), show that \(x \in H\). (This exercise is referred to in Chapter 25.)

Without looking at inner automorphisms of \(D_n\), determine the number of such automorphisms.

If N is a normal subgroup of G and |G/N| = m, show that \(x^m \in N\) for all x in G.

Suppose that a group G has a subgroup of order n. Prove that the intersection of all subgroups of G of order n is a normal subgroup of G.

Suppose that H is a normal subgroup of a finite group G. If G/H has an element of order n, show that G has an element of order n. Show, by example, that the assumption that G is finite is necessary.

Problem 65E If G is non-Abelian, show that Aut(G) is not cyclic.

Problem 62E Let G be a group and let G' be the subgroup of G generated by the set S = {x–1y–1 xy | x, y ? G}. (See Exercise 3, Supplementary Exercises for Chapters 5–8, for a more complete description of G'.) a. Prove that G' is normal in G. b. Prove that G/G' is Abelian. c. If G/N is Abelian, prove that G' ? N. d. Prove that if H is a subgroup of G and G'? H, then H is normal in G.

Let \(|G|=p^{n} m\), where p is prime and gcd(p, m) = 1. Suppose that H is a normal subgroup of G of order \(p^{n}\). If K is a subgroup of G of order \(p^{k}\), show that \(K \subseteq H\).

Recall that a subgroup N of a group G is called characteristic if \(\phi(N) = N\) for all automorphisms \(\phi\) of G. If N is a characteristic subgroup of G, show that N is a normal subgroup of G.

Problem 69E In D4, let Form an operation table for the cosets Is the result a group table? Does your answer contradict Theorem 9.2?

Prove that \(A_4\) is the only subgroup of \(S_4\) of order 12.

If H is a normal subgroup of G and |H| = 2, prove that H is contained in the center of G.

Problem 71E If |G| = 30 and |Z(G)| = 5, what is the structure of G/Z(G)?

Let G be a finite group and let H be an odd-order subgroup of G of index 2. Show that the product of all the elements of G (taken in any order) cannot belong to H.

Problem 75E Let G be a group and p a prime. Suppose that H = {gp Z g ? G} is a subgroup of G. Show that H is normal and that every nonidentity element of G/H has order p.

Prove that \(A_5\) cannot have a normal subgroup of order 2.

Problem 76E Suppose that H is a normal subgroup of G. If |H| = 4 and gH has order 3 in G/H, find a subgroup of order 12 in G.

Problem 77E Let G be a group and H an odd-order subgroup of G of index 2. Show that H contains every element of G of odd order.

A proper subgroup H of a group G is called maximal if there is no subgroup K such that \(H \subset K \subset G\) (that is, there is no subgroup K properly contained between H and $G)$. Show that Z(G) is never a maximal subgroup of a group G.