91Ó°ÊÓ

In Exercises 1 through 5 determine which of the indicated functions is a permutation of the indicated set. $$ f: \mathbb{R} \rightarrow \mathbb{R}, \text { where } f(x)=3 x+\sqrt{2} $$

Find the orders of the indicated elements in the indicated groups: (a) \(6 \in \mathbb{Z}_{10}\) (b) \(6 \in \mathbb{Z}_{15}\) (c) \(10 \in \mathbb{Z}_{42}\) (d) \(77 \in \mathbb{Z}_{210}\) (e) \(40 \in \mathbb{Z}_{210}\) (f) \(70 \in \mathbb{Z}_{210}\)

In Exercises 1 through 10 find the order of the indicated element in the indicated group. $$ 2 \in \mathbb{Z}_{3} $$

Show that the indicated set \(G\) with the specified operation forms a group by showing that the four axioms in the definition of a group are satisfied. \(G=2 Z\) under addition

In Exercises 1 through 5 determine which of the indicated functions is a permutation of the indicated set. $$ f: \mathbb{R} \rightarrow \mathbb{R}, \text { where } f(x)=3 x^{2}+2 $$

Find the order of the indicated element in the indicated group. $$ 4 \in \mathbb{Z}_{10} $$

Let \(G\) be a group and \(a \in G\) an element of order \(|a|=6\). (a) Write all the elements of \(\langle a\rangle\). (b) Find in \(\langle a\rangle\) the elements \(a^{32}, a^{47}, a^{70}\).

In Exercises 1 through 5 determine which of the indicated functions is a permutation of the indicated set. $$ f: U(5) \rightarrow U(5), \text { where } f(x)=x^{-1} $$

Show that the indicated set \(G\) with the specified operation forms a group by showing that the four axioms in the definition of a group are satisfied. \(G=C^{*}=C-\\{0\\}\) under complex multiplication

Find all the generators of \(\mathbb{Z}_{10}, \mathbb{Z}_{12},\) and \(\mathbb{Z}_{15}\).