Two elements \(\alpha\), \(\beta\) of a finite group \(G\) are called conjugate if there exists \(\gamma \in G\) such that \(\alpha = \gamma\beta\gamma^{-1}\) Show that conjugacy defines an equivalence relation. The elements \(a\), \(b\) have associative multiplication with unit \(e\) and satisfy \(a^3 = b^2 = (ab)^2 = e\) The set of six elements \(e\), \(a\), \(a^2\), \(b\), \(ab\), \(a^2b\) are distinct. Show that they form a group and separate them into equivalence classes under conjugacy.
The process of representing polynomials by their remainders upon division by \(x^2 + 1\) separates the set of all polynomials with real coefficients into equivalence classes. Denote by \((\alpha, \beta)\) the class containing \(\alpha x + \beta\). If the product \((\alpha_1, \beta_1)(\alpha_2, \beta_2)\) denotes the class containing the products of polynomials from \((\alpha_1, \beta_1)\) and \((\alpha_2, \beta_2)\), obtain it explicitly in the form \((\alpha, \beta)\). Specify the relationship of the field formed by the set of all \((\alpha, \beta)\) and the field of complex numbers. In a like procedure using remainders upon division by \(x^2 + x + 1\), denote by \([\lambda, \mu]\) the class containing \(\lambda x + \mu\). Evaluate the product \([\lambda_1, \mu_1][\lambda_2, \mu_2]\). By relating the set of all \([\lambda, \mu]\) to the complex numbers obtain a 1:1 correspondence between the sets \((\alpha, \beta)\) and \([\lambda, \mu]\) which makes explicit the fact that the corresponding product laws are isomorphic.
Show that the geometric mean of \(n\) positive numbers is less than or equal to their arithmetic mean. Use this result and the binomial theorem to show that \[(n+1)! \leq 2^n\left [1! \cdot 2! \cdots n! \right ]^{2/(n+1)}\]
Solution: \begin{align*} && 2^n &= \sum_{k=0}^n \binom{n}{k} \\ &&& \geq (n+1) \sqrt[n+1]{\prod_{k=0}^n \binom{n}{k}} \\ &&& = (n+1) \sqrt[n+1]{\frac{n!}{0! \cdot n!}\cdot\frac{n!}{1! \cdot (n-1)!}\cdots \frac{n!}{n! \cdot 0!}} \\ &&&=(n+1)^{n+1} \sqrt[n+1]{\frac{(n!)^{n+1}}{\left ( 0! \cdot 1! \cdots n! \right)^2}} \\ &&&= \frac{(n+1)!}{\left [1! \cdot 2! \cdots n! \right ]^{2/(n+1)}} \\ \Rightarrow && (n+1)! & \leq 2^n\left [1! \cdot 2! \cdots n! \right ]^{2/(n+1)} \end{align*}
Show that $\begin{vmatrix} 1+x_1 & x_2 & x_3 & \cdots & x_n \\ x_1 & 1+x_2 & x_3 & \cdots & x_n \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ x_1 & x_2 & x_3 & \cdots & 1+x_n \end{vmatrix} = 1 + x_1 + x_2 + \cdots + x_n$ Hence or otherwise evaluate the \(n\)-rowed determinant $\begin{vmatrix} 0 & 1 & 1 & \cdots & 1 \\ 1 & 0 & 1 & \cdots & 1 \\ 1 & 1 & 0 & \cdots & 1 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & 1 & 1 & \cdots & 0 \end{vmatrix}$
Find the two values of \(\lambda\) for which the matrix equation \(\begin{pmatrix} 3 & 2 \\ 1 & 4 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \lambda \begin{pmatrix} x \\ y \end{pmatrix}\) has non-trivial solutions for \(x\) and \(y\). For each of these values, find a corresponding solution for \(x\) and \(y\). If \(A = \begin{pmatrix} 3 & 2 \\ 1 & 4 \end{pmatrix}, \quad M = \begin{pmatrix} x_1 & x_2 \\ y_1 & y_2 \end{pmatrix}\) where \((x_1, y_1)\) and \((x_2, y_2)\) are the two solutions just obtained, find a diagonal matrix \(D\) such that \(AM = MD\)
\(A\), \(B\), \(C\), \(D\), \(E\), \(F\), \(G\) are consecutive vertices of a regular polygon of \(n\) sides (\(n \geq 7\)); \(BE\) meets \(DG\) in \(X\), \(CF\) in \(Y\), and \(AD\) in \(Z\). Prove that \(EX \cdot EZ = EY^2\).
A square \(ABCD\) is such that \(A\) lies on \(y = 0\), \(C\) on \(x = 0\), while \(B\) and \(D\) lie on the circle \(x^2 + y^2 + 2gx + 2fy = 0\) (\(f\), \(g\), \(f \neq g\) all non-zero). Show that three squares satisfy these conditions, and that any pair of these have a common vertex.
Prove that the normals to a parabola at the points \(Q\), \(R\) intersect on the curve if and only if \(QR\) passes through a certain fixed point. Suppose that this condition is satisfied, and let the normals at \(Q\), \(R\) meet at \(P\). If \(P'\) is the intersection of the parabola with the line parallel to the axis passing through the common point of \(QR\) and the directrix, show that \(PP'\) passes through the focus.
Interpret the equation \(S + \lambda T^2 = 0\), where \(S = 0\) and \(T = 0\) are the equations of a conic and one of its tangents, and \(\lambda\) is a constant. Hence or otherwise find the equations of the circles of curvature of the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) at the ends of the major axis. Show that these circles touch each other if \(a^2 = 2b^2\), and find the condition that each should be touched by the circle with the minor axis as diameter.
Two regular tetrahedra are formed from among the vertices of a cube of edge length \(a\). Find the volume of the portion of the cube external to both tetrahedra.