Final answer:
An infinite group where every element has a finite order includes the Prüfer p-group and the group of all roots of unity in the complex numbers. The Prüfer p-group is infinite, and each element has an order that is a power of a prime. The roots of unity group is infinite as well, with each element having an order that is a divisor of a positive integer.
Step-by-step explanation:
An example of an infinite group in which every element has a finite order is the group Prüfer p-group, also known as the quasicyclic group. In this group, each element apart from the identity has an order that is a power of a prime number p, and yet the group itself is infinite. This group is denoted by Z(pⁿ), where p is a prime.
Another example is the group of all roots of unity in the complex numbers. The roots of unity are the complex solutions to the equation x^n = 1 for some positive integer n. Every root of unity has a finite order (specifically, the order is a divisor of n), but since there are roots of unity for every positive integer n, there are infinitely many roots of unity, and thus they form an infinite group under multiplication.
An example of an infinite group in which every element has a finite order is the group of integers under addition modulo n, denoted as Z/nZ.
In this group, each element has a finite order because the group operation is defined as adding integers modulo n. For example, in Z/4Z, the element 2 has a finite order of 2 since 2 + 2 = 0 (mod 4). Similarly, the element 3 has a finite order of 4 since 3 + 3 + 3 + 3 = 0 (mod 4).
This group is infinite as it contains an infinite number of elements, but every element in the group has a finite order.