91Ó°ÊÓ

Let \(f: X \rightarrow Y\) and \(g: Y \rightarrow Z\) be given functions. Show that if \(f\) and \(g\) are injective, then so is \(g f: X \rightarrow Z\). Show also that if \(f\) and \(g\) are surjective, then so is \(g f: X \rightarrow Z\).

Show that any subgroup of \(S_{n}\) (that is, a subset of \(S_{n}\) that is a group in its own right) which is not contained in \(A_{n}\) contains an equal number of even and odd permutations.

Suppose that the permutation \(\rho\) of \(\\{1, \ldots, n\\}\) satisfies \(\rho^{3}=1\). Show that \(\rho\) is a product of 3 cycles, and deduce that if \(n\) is not divisible by 3 then \(\rho\) fixes some \(k\) in \(\\{1, \ldots, n\\}\).

Consider the (infinite) set \(\mathbb{Z}\) of integers. Show that there is a function \(f: \mathbb{Z} \rightarrow \mathbb{Z}\) that is injective but not surjective, and a function \(g: \mathbb{Z} \rightarrow \mathbb{Z}\) that is surjective but not injective. Now let \(X\) be any finite set, and let \(f: X \rightarrow X\) be any function. Show that the following statements are equivalent: (a) \(f: X \rightarrow X\) is injective; (b) \(f: X \rightarrow X\) is surjective; (c) \(f: X \rightarrow X\) is bijective.

Let \(G=\\{x \in \mathbb{R}: x \neq-1\\}\), where \(\mathbb{R}\) is the set of real numbers, and let \(x * y=x+y+x y\), where \(x y\) denotes the usual product of two real numbers. Show that \(G\) with the operation \(*\) is a group. What is the inverse \(2^{-1}\) of 2 in this group? Find \(\left(2^{-1}\right) * 6 *\left(5^{-1}\right)\), and hence solve the equation \(2 * x * 5=6\)