A group G is called cyclic if there is an element x, in G, such that every eleme

Question

A group G is called cyclic if there is an element x, in G, such that every element of G has the form xk for some fixed element of x element of G. Here Xk = x*x****x the product of x copies of x, when k is a positive integer; xk = (x-1)(absolute value of k) when k is a negative integer and x0 = e. Give an example of a cyclic group with infinitely many elements (that is, a cyclic group of infinite order). Give an example of a cyclic group with finitely many elements (that is, a cyclic group of finite order)

Explanation / Answer

A cyclic group of infinite order is the set of integers under addition, (Z, +). It can be generated by 1 or -1; indeed, every nonzero integer is of the form 1 + 1 + ... + 1 or (-1) + (-1) + ... + (-1). This is the only infinite cyclic group, if you find another infinite cyclic group, it will be isomorphic to (Z, +).

There are many cyclic groups of finite order, for example Z4 - the set of integers {0, 1, 2, 3} with addition modulo 4. It can be generated by 1 (trivially) and also by 3:

3 + 3 = 2

2 + 3 = 1

1 + 3 = 0

0 + 3 = 3

(here by "+" I mean "+mod4" of course)

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