Exercises 8.4 Exercises

Write out Cayley tables for groups formed by the symmetries of a rectangle and for \({\mathbb Z}_4^+\text{.}\) How many elements are in each group? Are the groups the isomorphic? Why or why not?

Describe the symmetries of a non-square rhombus and prove that the set of symmetries forms a group. Give Cayley tables for both the symmetries of a non-square rectangle and the symmetries of a non-square rhombus. Are the symmetries of a rectangle and those of a rhombus isomorphic?

Describe the symmetries of a square and prove that the set of symmetries is a group. Give a Cayley table for the symmetries. How many ways can the vertices of a square be permuted? Is each permutation necessarily a symmetry of the square? The symmetry group of the square is denoted by \(D_4\text{.}\)

Let \(G\) be the group of symmetries of an equilateral triangle as in Figure 8.2. Find a symmetry \(x\) such that \(\rho_1 x \rho_1 = \mu_1\text{.}\)

Give a multiplication table for the group \(\mathbb Z_{12}^\times\text{.}\)

Let \(S = {\mathbb R} \setminus \{ -1 \}\) and define a binary operation on \(S\) by \(a \ast b = a + b + ab\text{.}\) Prove that \((S, \ast)\) is an abelian group.

Give an example of two elements \(A\) and \(B\) in \(GL_2({\mathbb R})\) with \(AB \neq BA\text{.}\)

Prove that the product of two matrices in \(SL_2({\mathbb R})\) has determinant one.

Prove that \(\det(AB) = \det(A) \det(B)\) in \(GL_2({\mathbb R})\text{.}\) Use this result to show that the binary operation in the group \(GL_2({\mathbb R})\) is closed; that is, if \(A\) and \(B\) are in \(GL_2({\mathbb R})\text{,}\) then \(AB \in GL_2({\mathbb R})\text{.}\)

Show that \({\mathbb R}^{\times} = {\mathbb R} \setminus \{0 \}\) is a group under the operation of multiplication.

Given the groups \({\mathbb R}^{\times}\) and \({\mathbb Z}^+\text{,}\) let \(G = {\mathbb R}^{\times} \times {\mathbb Z}^+\text{.}\) Define a binary operation \(\circ\) on \(G\) by \((a,m) \circ (b,n) = (ab, m + n)\text{.}\) Show that \(G\) is a group under this operation.

Prove or disprove that every group containing six elements is abelian.

Give a specific example of some group \(G\) and elements \(g, h \in G\) where \((gh)^n \neq g^nh^n\text{.}\)

Give an example of three different groups with eight elements. Why are the groups different?

Show that there are \(n!\) permutations of a set containing \(n\) items.

Let \(a\) and \(b\) be elements in a group \(G\text{.}\) Prove that \(ab^na^{-1} = (aba^{-1})^n\) for \(n \in \mathbb Z\text{.}\)

Let \(\mathbb Z_n^\times\) be the group of units. If \(n \gt 2\text{,}\) prove that there is an element \(k \in \mathbb Z_n^\times\) such that \(k^2 = 1\) and \(k \neq 1\text{.}\)

Prove that the inverse of \(g _1 g_2 \cdots g_n\) is \(g_n^{-1} g_{n-1}^{-1} \cdots g_1^{-1}\text{.}\)

Prove the remainder of Proposition 8.20: if \(G\) is a group and \(a, b \in G\text{,}\) then the equation \(xa = b\) has a unique solution in \(G\text{.}\)

Prove Theorem 8.22.

Prove the right and left cancellation laws for a group \(G\text{;}\) that is, show that in the group \(G\text{,}\) \(ba = ca\) implies \(b = c\) and \(ab = ac\) implies \(b = c\) for elements \(a, b, c \in G\text{.}\)

Show that if \(a^2 = e\) for all elements \(a\) in a group \(G\text{,}\) then \(G\) must be abelian.

Show that if \(G\) is a finite group of even order, then there is an \(a \in G\) such that \(a\) is not the identity and \(a^2 = e\text{.}\)

Let \(G\) be a group and suppose that \((ab)^2 = a^2b^2\) for all \(a\) and \(b\) in \(G\text{.}\) Prove that \(G\) is an abelian group.

Find all the subgroups of \({\mathbb Z}_3^+ \times {\mathbb Z}_3^+\text{.}\) Use this information to show that \({\mathbb Z}_3^+ \times {\mathbb Z}_3^+\) is not isomorphic to \({\mathbb Z}_9^+\text{.}\) (See Example 8.27 for a short description of the product of groups.)

Find all the subgroups of the symmetry group of an equilateral triangle.

Compute the subgroups of the symmetry group of a square.

Let \(H = \{2^k : k \in {\mathbb Z} \}\text{.}\) Show that \(H\) is a subgroup of \({\mathbb Q}^\times\text{.}\)

Let \(n = 0, 1, 2, \ldots\) and \(n {\mathbb Z} = \{ nk : k \in {\mathbb Z} \}\text{.}\) Prove that \(n {\mathbb Z}\) is a subgroup of \({\mathbb Z^+}\text{.}\) Show that these subgroups are the only subgroups of \(\mathbb{Z}^+\text{.}\)

Let \({\mathbb T} = \{ z \in {\mathbb C}^\times : |z| =1 \}\text{.}\) Prove that \({\mathbb T}\) is a subgroup of \({\mathbb C}^\times\text{.}\)

Prove or disprove: \(SL_2( {\mathbb Z} )\text{,}\) the set of \(2 \times 2\) matrices with integer entries and determinant one, is a subgroup of \(SL_2( {\mathbb R} )\text{.}\)

Prove or disprove: \(H\text{,}\) the set of \(2 \times 2\) matrices with integer entries and non-zero determinant is a subgroup of \(GL_2(\mathbb R)\text{.}\)

List the subgroups of the quaternion group, \(Q_8\text{.}\)

Prove that the intersection of two subgroups of a group \(G\) is also a subgroup of \(G\text{.}\)

Prove or disprove: If \(H\) and \(K\) are subgroups of a group \(G\text{,}\) then \(H \cup K\) is a subgroup of \(G\text{.}\)

Prove or disprove: If \(H\) and \(K\) are subgroups of a group \(G\text{,}\) then \(H K = \{hk : h \in H \text{ and } k \in K \}\) is a subgroup of \(G\text{.}\) What if \(G\) is abelian?

Let \(a\) and \(b\) be elements of a group \(G\text{.}\) If \(a^4 b = ba\) and \(a^3 = e\text{,}\) prove that \(ab = ba\text{.}\)

Let \(a\) and \(b\) be elements of a group \(G\text{.}\) If \(a^2=e\) and \((ab)^2 = e\text{,}\) then prove that \((ba)^2 = e\text{.}\)

Give an example of an infinite group in which every nontrivial subgroup is infinite.

If \(xy = x^{-1} y^{-1}\) for all \(x\) and \(y\) in \(G\text{,}\) prove that \(G\) must be abelian.

Prove or disprove: Every proper subgroup of an nonabelian group is nonabelian.

Let \(H\) be a subgroup of \(G\text{.}\) If \(g \in G\text{,}\) show that \(gHg^{-1} = \{ghg^{-1} : h\in H\}\) is also a subgroup of \(G\text{.}\)