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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 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.

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

\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 \(n \in \mathbb Z\text{.}\)

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 2.21: 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 2.23.

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.

These exercises are about becoming comfortable working with groups in Sage. Sage worksheets have extensive capabilities for making new cells with carefully formatted text, include support for LaTeX syntax to express mathematics. So when a question asks for explanation or commentary, make a new cell and communicate clearly with your audience.

Create the groups `CyclicPermutationGroup(8)` and `DihedralGroup(4)` and name these groups `C` and `D`, respectively. We will understand these constructions better shortly, but for now just understand that both objects you create are actually groups.

Check that `C` and `D` have the same size by using the `.order()` method. Determine which group is abelian, and which is not, by using the `.is_abelian()` method.

Use the `.cayley_table()` method to create the Cayley table for each group.

Write a nicely formatted discussion identifying differences between the two groups that are discernible in properties of their Cayley tables. In other words, what is {\em different} about these two groups that you can “see” in the Cayley tables? (In the Sage notebook, a Shift-click on a blue bar will bring up a mini-word-processor, and you can use use dollar signs to embed mathematics formatted using TeX syntax.)