Chapter 7

Let \(A\) be a set and \(a \in A\). Let \(G\) be the subset of \(S_{A}\) consisting of all the permutations \(f\) of \(A\) such that \(f(a)=a\). Prove that \(G\) is a subgroup of \(S_{A}\).

Consider the polynomial \(p=\left(x_{1}-x_{2}\right)^{2}+\left(x_{3}-x_{4}\right)^{2}\). It is unaltered when the subscripts undergo any of the following permutations: $$ \begin{array}{lll} \left(\begin{array}{llll} 1 & 2 & 3 & 4 \\ 2 & 1 & 3 & 4 \end{array}\right) & \left(\begin{array}{llll} 1 & 2 & 3 & 4 \\ 1 & 2 & 4 & 3 \end{array}\right) & \left(\begin{array}{llll} 1 & 2 & 3 & 4 \\ 2 & 1 & 4 & 3 \end{array}\right) & \left(\begin{array}{llll} 1 & 2 & 3 & 4 \\ 3 & 4 & 1 & 2 \end{array}\right) \\ \left(\begin{array}{llll} 1 & 2 & 3 & 4 \\ 4 & 3 & 1 & 2 \end{array}\right) & \left(\begin{array}{llll} 1 & 2 & 3 & 4 \\ 3 & 4 & 2 & 1 \end{array}\right) & \left(\begin{array}{llll} 1 & 2 & 3 & 4 \\ 4 & 3 & 2 & 1 \end{array}\right) & \left(\begin{array}{llll} 1 & 2 & 3 & 4 \\ 1 & 2 & 3 & 4 \end{array}\right) \end{array} $$For example, the first of these permutations replaces \(p\) by $$ \left(x_{2}-x_{1}\right)^{2}+\left(x_{3}-x_{4}\right)^{2} $$ the second permutation replaces \(p\) by \(\left(x_{1}-x_{2}\right)^{2}+\left(x_{4}-x_{3}\right)^{2} ;\) and so on. The symmetries of a polynomial \(p\) are all the permutations of the subscripts which leave \(p\) unchanged. They form a group of permutations. List the symmetries of each of the following polynomials, and write their group table. $$ p=x_{1} x_{2}+x_{2} x_{3} $$

For any pair of real numbers \(a \neq 0\) and \(b\), define a function \(f_{a, b}\) as follows: $$ f_{a, b}(x)=a x+b $$ Prove that \(f_{a, b}\) is a permutation of \(\mathbb{R}\), that is, \(f_{a, b} \in S_{R}\).

For each integer \(n\), define \(f_{n}\) by: \(f_{n}(x)=x+n .\) Prove the following: For each integer \(n, f_{n}\) is a permutation of \(\mathbb{R}\), that is, \(f_{n} \in S_{\mathrm{H}}\)

In each of the following, \(A\) is a subset of \(\mathbb{R}\) and \(G\) is a set of permutations of \(A\). Show that \(G\) is a subgroup of \(S_{A}\), and write the table of \(G\). \(A\) is the set of all \(x \in \mathbb{R}\) such that \(x \neq 0,1 . G=\\{\varepsilon, f, g\\}\), where \(f(x)=1 /(1-x)\) and \(g(x)=(x-1) / x\).

Let \(G\) be the subset of \(S_{4}\) consisting of the permutations $$ \begin{array}{ll} \varepsilon=\left(\begin{array}{llll} 1 & 2 & 3 & 4 \\ 1 & 2 & 3 & 4 \end{array}\right) & f=\left(\begin{array}{llll} 1 & 2 & 3 & 4 \\ 2 & 1 & 4 & 3 \end{array}\right) \\ g=\left(\begin{array}{llll} 1 & 2 & 3 & 4 \\ 3 & 4 & 1 & 2 \end{array}\right) & h=\left(\begin{array}{llll} 1 & 2 & 3 & 4 \\ 4 & 3 & 2 & 1 \end{array}\right) \end{array} $$ Show that \(G\) is a group of permutations, and write its table. \begin{tabular}{l|llll} & \(\varepsilon\) & \(f\) & \(g\) & \(h\) \\ \hline\(\varepsilon\) & & & \\ \(f\) & & & \\ \(g\) & & & \\ \(h\) & & & \end{tabular}

Consider the following permutations \(f, g\), and \(h\) in \(S_{6}\) : $$ \begin{gathered} f=\left(\begin{array}{llllll} 1 & 2 & 3 & 4 & 5 & 6 \\ 6 & 1 & 3 & 5 & 4 & 2 \end{array}\right) \quad g=\left(\begin{array}{llllll} 1 & 2 & 3 & 4 & 5 & 6 \\ 2 & 3 & 1 & 6 & 5 & 4 \end{array}\right) \\ h=\left(\begin{array}{llllll} 1 & 2 & 3 & 4 & 5 & 6 \\ 5 & 1 & 6 & 4 & 5 & 2 \end{array}\right) \end{gathered} $$ Compute the following: \(f^{-1}=\left(\begin{array}{cccccc}1 & 2 & 3 & 4 & 5 & 6 \\ & & & & & \end{array}\right) \quad g^{-1}=\left(\begin{array}{cccccc}1 & 2 & 3 & 4 & 5 & 6 \\ & & & & & \end{array}\right)\) \(h^{-1}=\left(\begin{array}{llllll}1 & 2 & 3 & 4 & 5 & 6 \\ & & & & & \end{array}\right)\) \(f \circ g=\left(\begin{array}{cccccc}1 & 2 & 3 & 4 & 5 & 6 \\ \vdots & & & & \end{array}\right) \quad g \circ f=\left(\begin{array}{llllll}1 & 2 & 3 & 4 & 5 & 6\end{array}\right)\)

If \(f\) is a permutation of \(A\) and \(a \in A\), we say that \(f\) moves \(a\) if \(f(a) \neq a\). Let \(A\) be an infinite set, and let \(G\) be the subset of \(S_{A}\) which consists of all the permutations \(f\) of \(A\) which move only a finite number of elements of \(A\). Prove that \(G\) is a subgroup of \(S_{A}\).

Consider the polynomial \(p=\left(x_{1}-x_{2}\right)^{2}+\left(x_{3}-x_{4}\right)^{2}\). It is unaltered when the subscripts undergo any of the following permutations: $$ \begin{array}{lll} \left(\begin{array}{llll} 1 & 2 & 3 & 4 \\ 2 & 1 & 3 & 4 \end{array}\right) & \left(\begin{array}{llll} 1 & 2 & 3 & 4 \\ 1 & 2 & 4 & 3 \end{array}\right) & \left(\begin{array}{llll} 1 & 2 & 3 & 4 \\ 2 & 1 & 4 & 3 \end{array}\right) & \left(\begin{array}{llll} 1 & 2 & 3 & 4 \\ 3 & 4 & 1 & 2 \end{array}\right) \\ \left(\begin{array}{llll} 1 & 2 & 3 & 4 \\ 4 & 3 & 1 & 2 \end{array}\right) & \left(\begin{array}{llll} 1 & 2 & 3 & 4 \\ 3 & 4 & 2 & 1 \end{array}\right) & \left(\begin{array}{llll} 1 & 2 & 3 & 4 \\ 4 & 3 & 2 & 1 \end{array}\right) & \left(\begin{array}{llll} 1 & 2 & 3 & 4 \\ 1 & 2 & 3 & 4 \end{array}\right) \end{array} $$For example, the first of these permutations replaces \(p\) by $$ \left(x_{2}-x_{1}\right)^{2}+\left(x_{3}-x_{4}\right)^{2} $$ the second permutation replaces \(p\) by \(\left(x_{1}-x_{2}\right)^{2}+\left(x_{4}-x_{3}\right)^{2} ;\) and so on. The symmetries of a polynomial \(p\) are all the permutations of the subscripts which leave \(p\) unchanged. They form a group of permutations. List the symmetries of each of the following polynomials, and write their group table. $$ p=\left(x_{1}-x_{2}\right)\left(x_{2}-x_{3}\right)\left(x_{1}-x_{3}\right) $$

Let \(G\) be the group of symmetries of the rectangle. List the elements of \(G\) (there are four of them), and write the table of \(G\).

For any pair of real numbers \(a \neq 0\) and \(b\), define a function \(f_{a, b}\) as follows: $$ f_{a, b}(x)=a x+b $$ Prove that \(f_{a, b} \circ f_{c, d}=f_{\text {ac, ad }+b}\).

For each integer \(n\), define \(f_{n}\) by: \(f_{n}(x)=x+n .\) Prove the following: $$ f_{n} \circ f_{m}=f_{n+m} \quad \text { and } \quad f_{n}^{-1}=f_{-n} $$

List the elements of the cyclic subgroup of \(S_{6}\) generated by $$ f=\left(\begin{array}{llllll} 1 & 2 & 3 & 4 & 5 & 6 \\ 2 & 3 & 4 & 1 & 6 & 5 \end{array}\right) $$

Let \(A\) be a finite set, and \(B\) a subset of \(A\). Let \(G\) be the subset of \(S_{A}\) consisting of all the permutations \(f\) of \(A\) such that \(f(x) \in B\) for every \(x \in B\). Prove that \(G\) is a subgroup of \(S_{A}\).

Consider the polynomial \(p=\left(x_{1}-x_{2}\right)^{2}+\left(x_{3}-x_{4}\right)^{2}\). It is unaltered when the subscripts undergo any of the following permutations: $$ \begin{array}{lll} \left(\begin{array}{llll} 1 & 2 & 3 & 4 \\ 2 & 1 & 3 & 4 \end{array}\right) & \left(\begin{array}{llll} 1 & 2 & 3 & 4 \\ 1 & 2 & 4 & 3 \end{array}\right) & \left(\begin{array}{llll} 1 & 2 & 3 & 4 \\ 2 & 1 & 4 & 3 \end{array}\right) & \left(\begin{array}{llll} 1 & 2 & 3 & 4 \\ 3 & 4 & 1 & 2 \end{array}\right) \\ \left(\begin{array}{llll} 1 & 2 & 3 & 4 \\ 4 & 3 & 1 & 2 \end{array}\right) & \left(\begin{array}{llll} 1 & 2 & 3 & 4 \\ 3 & 4 & 2 & 1 \end{array}\right) & \left(\begin{array}{llll} 1 & 2 & 3 & 4 \\ 4 & 3 & 2 & 1 \end{array}\right) & \left(\begin{array}{llll} 1 & 2 & 3 & 4 \\ 1 & 2 & 3 & 4 \end{array}\right) \end{array} $$For example, the first of these permutations replaces \(p\) by $$ \left(x_{2}-x_{1}\right)^{2}+\left(x_{3}-x_{4}\right)^{2} $$ the second permutation replaces \(p\) by \(\left(x_{1}-x_{2}\right)^{2}+\left(x_{4}-x_{3}\right)^{2} ;\) and so on. The symmetries of a polynomial \(p\) are all the permutations of the subscripts which leave \(p\) unchanged. They form a group of permutations. List the symmetries of each of the following polynomials, and write their group table. $$ p=x_{1} x_{2}+x_{2} x_{3}+x_{1} x_{3} $$

List the symmetries of the letter \(\mathbf{Z}\) and give the table of this group of symmetries. Do the same for the letters \(\mathbf{V}\) and \(\mathbf{H}\).

Find a four-element abelian subgroup of \(S_{4}\). Write its table.

Consider the polynomial \(p=\left(x_{1}-x_{2}\right)^{2}+\left(x_{3}-x_{4}\right)^{2}\). It is unaltered when the subscripts undergo any of the following permutations: $$ \begin{array}{lll} \left(\begin{array}{llll} 1 & 2 & 3 & 4 \\ 2 & 1 & 3 & 4 \end{array}\right) & \left(\begin{array}{llll} 1 & 2 & 3 & 4 \\ 1 & 2 & 4 & 3 \end{array}\right) & \left(\begin{array}{llll} 1 & 2 & 3 & 4 \\ 2 & 1 & 4 & 3 \end{array}\right) & \left(\begin{array}{llll} 1 & 2 & 3 & 4 \\ 3 & 4 & 1 & 2 \end{array}\right) \\ \left(\begin{array}{llll} 1 & 2 & 3 & 4 \\ 4 & 3 & 1 & 2 \end{array}\right) & \left(\begin{array}{llll} 1 & 2 & 3 & 4 \\ 3 & 4 & 2 & 1 \end{array}\right) & \left(\begin{array}{llll} 1 & 2 & 3 & 4 \\ 4 & 3 & 2 & 1 \end{array}\right) & \left(\begin{array}{llll} 1 & 2 & 3 & 4 \\ 1 & 2 & 3 & 4 \end{array}\right) \end{array} $$For example, the first of these permutations replaces \(p\) by $$ \left(x_{2}-x_{1}\right)^{2}+\left(x_{3}-x_{4}\right)^{2} $$ the second permutation replaces \(p\) by \(\left(x_{1}-x_{2}\right)^{2}+\left(x_{4}-x_{3}\right)^{2} ;\) and so on. The symmetries of a polynomial \(p\) are all the permutations of the subscripts which leave \(p\) unchanged. They form a group of permutations. List the symmetries of each of the following polynomials, and write their group table. $$ p=\left(x_{1}-x_{2}\right)\left(x_{3}-x_{4}\right) $$

For any pair of real numbers \(a \neq 0\) and \(b\), define a function \(f_{a, b}\) as follows: $$ f_{a, b}(x)=a x+b $$ Let \(G=\left\\{f_{a, b}: a \in \mathbb{R}, b \in \mathbb{R}, a \neq 0\right\\} .\) Show that \(G\) is a subgroup of \(S_{R}\)

In each of the following, \(A\) is a subset of \(\mathbb{R}\) and \(G\) is a set of permutations of \(A\). Show that \(G\) is a subgroup of \(S_{A}\), and write the table of \(G\). \(A\) is the set of all the real numbers \(x \neq 0,1,2 . G\) is the subgroup of \(S_{A}\) generated by \(f(x)=2-x\) and \(g(x)=2 / x .(G\) has eight elements. List them, and write the table of \(G .\).)