Abstract Algebra: Theory and Applications

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 same? Why or why not?

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

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 \(D_4\text{.}\)

Give a multiplication table for the group \(U(12)\text{.}\)

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

Given the groups \({\mathbb R}^{\ast}\) and \({\mathbb Z}\text{,}\) let \(G = {\mathbb R}^{\ast} \times {\mathbb Z}\text{.}\) 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.

Show that addition and multiplication mod \(n\) are well defined operations. That is, show that the operations do not depend on the choice of the representative from the equivalence classes mod \(n\text{.}\)

Show that addition and multiplication mod \(n\) are associative operations.

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 \(U(n)\) be the group of units in \({\mathbb Z}_n\text{.}\) If \(n \gt 2\text{,}\) prove that there is an element \(k \in U(n)\) 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 3.21: 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 3.23.

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 the same group as \({\mathbb Z}_9\text{.}\) (See Example 3.28 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 \({\mathbb Q}^*\text{.}\)

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

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 \(SL_2( {\mathbb R} )\text{.}\)

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 \(ab = ba\text{.}\)

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 a 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 \(G\text{.}\)