(adsbygoogle = window.adsbygoogle || []).push({}); 1. Show that every element of the quotient group G = Q/Z has finite

order. Does G have finite order?

he problem statement, all variables and given/known data

2. This is the proof

The cosets that make up Q/Z have the form Z + q,

where q belongs to Q. For example, there is a coset Z + 1/2, which is

the set of all numbers of the form {n + 1/2}, where n is an integer.

And there is a coset Z + 2/5, which consists of all numbers of the

form n + 2/5, where n is an integer. The cosets form a group if you

define the sum of A and B to be the set of all sums of an element in A

and an element in B, and this group is the quotient group. The

identity of Q/Z is just Z.

Now if you take a rational number r/s, where r and s are integers,

then

s (r/s) = r

which is an integer. Now anything in the coset Z + r/s is an integer

plus r/s, so if you multiply anything in that coset by s, you get an

integer. So if you multiply the coset by s (i.e. add it to itself s

times) you get a coset consisting of all integers, but that's just Z

itself. That is, the coset is of finite order s (or a divisor of s).

3.I just need it to be explained especially the part with s(r/s)

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# Homework Help: Quotient Groups

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