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Find all \(x \in {\mathbb Z}\) satisfying each of the following equations.

\(3x \equiv 2 \pmod{7}\)

\(5x + 1 \equiv 13 \pmod{23}\)

\(5x + 1 \equiv 13 \pmod{26}\)

\(9x \equiv 3 \pmod{5}\)

\(5x \equiv 1 \pmod{6}\)

\(3x \equiv 1 \pmod{6}\)

Find all \(x \in {\mathbb Z}\) satisfying each of the following equations.

\(3x \equiv 2 \pmod{7}\)

\(5x + 1 \equiv 13 \pmod{23}\)

\(5x + 1 \equiv 13 \pmod{26}\)

\(9x \equiv 3 \pmod{5}\)

\(5x \equiv 1 \pmod{6}\)

\(3x \equiv 1 \pmod{6}\)

Which of the following multiplication tables defined on the set \(G = \{ a, b, c, d \}\) form a group? Support your answer in each case.

- \begin{equation*} \begin{array}{c|cccc} \circ & a & b & c & d \\ \hline a & a & c & d & a \\ b & b & b & c & d \\ c & c & d & a & b \\ d & d & a & b & c \end{array} \end{equation*}
- \begin{equation*} \begin{array}{c|cccc} \circ & a & b & c & d \\ \hline a & a & b & c & d \\ b & b & a & d & c \\ c & c & d & a & b \\ d & d & c & b & a \end{array} \end{equation*}
- \begin{equation*} \begin{array}{c|cccc} \circ & a & b & c & d \\ \hline a & a & b & c & d \\ b & b & c & d & a \\ c & c & d & a & b \\ d & d & a & b & c \end{array} \end{equation*}
- \begin{equation*} \begin{array}{c|cccc} \circ & a & b & c & d \\ \hline a & a & b & c & d \\ b & b & a & c & d \\ c & c & b & a & d \\ d & d & d & b & c \end{array} \end{equation*}

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 \(U(12)\text{.}\)

HintLet \(S = {\mathbb R} \setminus \{ -1 \}\) 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 \(AB \neq BA\text{.}\)

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

Prove that the set of matrices of the form

\begin{equation*} \begin{pmatrix} 1 & x & y \\ 0 & 1 & z \\ 0 & 0 & 1 \end{pmatrix} \end{equation*}is a group under matrix multiplication. This group, known as the *Heisenberg group*, is important in quantum physics. Matrix multiplication in the Heisenberg group is defined by

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 \(AB \in GL_2({\mathbb R})\text{.}\)

Let \({\mathbb Z}_2^n = \{ (a_1, a_2, \ldots, a_n) : a_i \in {\mathbb Z}_2 \}\text{.}\) Define a binary operation on \({\mathbb Z}_2^n\) by

\begin{equation*} (a_1, a_2, \ldots, a_n) + (b_1, b_2, \ldots, b_n) = (a_1 + b_1, a_2 + b_2, \ldots, a_n + b_n). \end{equation*}Prove that \({\mathbb Z}_2^n\) is a group under this operation. This group is important in algebraic coding theory.

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.

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

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

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

HintShow that

\begin{equation*} 0 + a \equiv a + 0 \equiv a \pmod{ n } \end{equation*}for all \(a \in {\mathbb Z}_n\text{.}\)

Prove that there is a multiplicative identity for the integers modulo \(n\text{:}\)

\begin{equation*} a \cdot 1 \equiv a \pmod{n}. \end{equation*}For each \(a \in {\mathbb Z}_n\) find an element \(b \in {\mathbb Z}_n\) such that

\begin{equation*} a + b \equiv b + a \equiv 0 \pmod{ n}. \end{equation*}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.

Show that multiplication distributes over addition modulo \(n\text{:}\)

\begin{equation*} a(b + c) \equiv ab + ac \pmod{n}. \end{equation*}Let \(a\) and \(b\) be elements in a group \(G\text{.}\) Prove that \(ab^na^{-1} = (aba^{-1})^n\) for \(n \in \mathbb Z\text{.}\)

HintLet \(U(n)\) 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 Proposition 3.2.14: 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 Theorem 3.2.16.

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.

HintShow 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 Example 3.3.5 for a short description of the product of groups.)

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

HintCompute 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 \({\mathbb C}^*\text{.}\)

where \(\theta \in {\mathbb R}\text{.}\) Prove that \(G\) is a subgroup of \(SL_2({\mathbb R})\text{.}\)

Prove that

\begin{equation*} G = \{ a + b \sqrt{2} : a, b \in {\mathbb Q} \text{ and } a \text{ and } b \text{ are not both zero} \} \end{equation*}is a subgroup of \({\mathbb R}^{\ast}\) under the group operation of multiplication.

HintLet \(G\) be the group of \(2 \times 2\) matrices under addition and

\begin{equation*} H = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} : a + d = 0 \right\}. \end{equation*}Prove that \(H\) is a subgroup of \(G\text{.}\)

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 \(G\text{.}\)

HintProve 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 \(G\) be a group and \(g \in G\text{.}\) Show that

\begin{equation*} Z(G) = \{ x \in G : gx = xg \text{ for all } g \in G \} \end{equation*}is a subgroup of \(G\text{.}\) This subgroup is called the *center* of \(G\text{.}\)

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

HintGive 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 \(G\) and

\begin{equation*} C(H) = \{ g \in G : gh = hg \text{ for all } h \in H \}. \end{equation*}Prove \(C(H)\) is a subgroup of \(G\text{.}\) This subgroup is called the *centralizer* of \(H\) in \(G\text{.}\)

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