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**1. The problem statement, all variables and given/known data**

True or False? Every infinite group has an element of infinite order.

**2. Relevant equations**

A group is a set G along with an operation * such that

if a,b,c [itex]\in[/itex] G then

(a*b)*c=a*(b*c)

there exists an e in G such that a*e=a

for every a in G there exists an a' such that a*a'=e

The order of an element is the smallest number of times it needs to be operated with itself to become equal to the identity.

**3. The attempt at a solution**

The back of the book says this is false. But I am having a hard time thinking of an infinite group where every element has finite order. Perhaps maybe the group of integers under subtraction? This is indeed a group because for a,b in Z, a-b is in Z. And a-0=a. and a-a=0 so a is its own inverse.

So every element in Z has order 2 but the group is infinite because there are infinite integers ... Is this right?

Thanks