91Ó°ÊÓ

Give an example of two subgroups \(H\) and \(K\) of a group \(G\) whose union \(H \cup K\) is not a subgroup of \(G\).

If \(G\) is a group and \(G / Z(G)\) is cyclic, where \(Z(G)\) denotes the center of \(G\), prove that \(G\) is abelian; that is, \(G=Z(G)\). Conclude that if \(G\) is not abelian, then \(G / Z(G)\) is never cyclic.

(i) If \(1

(i) If \(\alpha\) is an \(r\)-cycle, show that \(\alpha^{r}=(1)\). (ii) If \(\alpha\) is an \(r\)-cycle, show that \(r\) is the smallest positive integer \(k\) such that \(\alpha^{k}=(1)\)

Prove that a group \(G\) is abelian if and only if the function \(f: G \rightarrow G\), given by \(f(a)=a^{-1}\), is a homomorphism.

This exercise gives some invariants of a group \(G .\) Let \(f: G \rightarrow H\) be an isomorphism. (i) Prove that if \(a \in G\) has infinite order, then so does \(f(a)\), and if \(a\) has finite order \(n\), then so does \(f(a)\). Conclude that if \(G\) has an element of some order \(n\) and \(H\) does not, then \(G \nsupseteq H\). (ii) Prove that if \(G \cong H\), then, for every divisor \(k\) of \(|G|\), both \(G\) and \(H\) have the same number of elements of order \(k\).

If \(p\) is a prime and \(G\) is a finite group in which every element has order a power of \(p\), prove that \(G\) is a \(p\)-group.

\text { Show that an } r \text {-cycle is an even permutation if and only if } r \text { is odd. }

The stochastic \(^{12}\) group \(\Sigma(2, \mathbb{R})\) consists of all those matrices in \(\mathrm{GL}(2, \mathrm{R})\) whose column sums are 1 ; that is, \(\Sigma(2, \mathbb{R})\) consists of all the nonsingular matrices \(\left[\begin{array}{l}a c \\ b d\end{array}\right]\) with \(a+b=1=c+d\). [There are also stochastic groups \(\Sigma(2, \mathbb{Q})\) and \(\Sigma(2, \mathbb{C}) .]\) Prove that the product of two stochastic matrices is again stochastic, and that the inverse of a stochastic matrix is stochastic.

(i) If \(H\) is a subgroup of \(G\) and if \(x \in H\), prove that $$ C_{H}(x)=H \cap C_{G}(x) $$ (ii) If \(H\) is a subgroup of index 2 in a finite group \(G\) and if \(x \in H\), prove that either \(\left|x^{H}\right|=\left|x^{G}\right|\) or \(\left|x^{H}\right|=\frac{1}{2}\left|x^{G}\right|\), where \(x^{H}\) is the conjugacy class of \(x\) in \(H\).