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