91Ó°ÊÓ

Let \(R^{*}\) be the multiplicative group of non-zero real numbers. Describe explicitly the kernel of the absolute value homomorphism $$ x-|x| $$ of \(\mathrm{R}^{*}\) into itself. What is the image of this homomorphism?

Using the structure theorem for abelian groups, prove the following: (a) An abelian group is cyclic if and only if, for every prime \(p\) dividing the order of \(G\), there is one and only one subgroup of order \(p\). (b) Let \(G\) be a finite abelian group which is not cyclic. Then there exists a prime \(p\) such that \(G\) contains a subgroup \(C_{1} \times C_{2}\) where \(C_{1}\) and \(C_{2}\) are cyclic of order \(p\).

Prove that every group of prime order is cyclic.

Let \(f: A \rightarrow B\) be a homomorphism of abelian groups. Assume that there exists a homomorphism \(g: B \rightarrow A\) such that \(f=g=\mathrm{id}_{B}\). (a) Prove that \(A\) is the direct sum $$ A=\text { Ker } f \oplus \operatorname{Im} g $$

Show that the number of odd permutations of \((1, \ldots, n)\) for \(n \geq 2\) is equal to the number of even permutations.

Show that the groups \(S_{2}, S_{3}, S_{4}\) are solvable. [Hint: For \(S_{4}\), find a subgroup \(H\) of order 4 in \(A_{4}\). Consider the homomorphism of \(A_{4}\) into \(S_{3}\) given by translation on the cosets of \(H\). Analyze the kernel of this homomorphism.]

Show that the group of inner automorphisms is normal in the group of all automorphisms of a group \(G\).

Two cycles \(\left[t_{1} \cdots i_{n}\right]\) and \(\left[j_{1} \cdots-j\right]\) are said to be disjoint if no integer \(i\), is equal to any integer \(J .\). Prove that a permutation is equal to a product of disjoint cycles.

Let \(G\) be a finite group of order \(2 k\) for some positive integer \(k\). (a) Prove that \(G\) has an element of period 2. [Hint: show that there exists \(x \in G . x \neq e .\) such that \(\left.x=x^{-1} .\right]\) (b) Assume that \(k\) is odd. Let \(a \in G\) have period 2 and let \(T_{a}: G \rightarrow G\) be translation by a. Prove that \(T_{a}\) is an odd permutation.

Let \(\varphi\) be the Euler phi function, i.e. \(\phi(n)\) is the order of \((\mathbf{Z} / n \mathbf{Z})^{*}\). (a) If \(p\) is a prime number, and \(r\) an integer \(\geq 1\), show that $$ \varphi\left(p^{\prime}\right)=(p-1) p^{\prime-1} $$ (b) Prove that if \(m, n\) are positive relatively prime integers, then $$ \varphi(m n)=\varphi(m) \rho(n) $$ (You may (should) of course use previous exercises to do this.)