Proof of Finite Order of G in Quotient Group Q/Z

In summary: sorry if i'm not making sense, just trying to be clear about why this proof still applies even when s is not prime
  • #1
Bellarosa
48
0
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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  • #2
What you said is perfectly correct. To put it in precise terms, if r/s+Z is an element of Q/Z (where we may assume that s is a positive integer), then s(r/s+Z)=r+Z=Z. So every element of Q/Z has finite order.
 
  • #3
ok... I can see by example why this proof make sense with s in r/s being a prime number because s here is the order of the factor group, how about when s is not prime, does the proof still apply?
 
  • #4
Why would s have to be a prime number? The order of what factor group? Q/Z is definitely not of order s, or any other finite number!
 
  • #5
ok...
 

1. What is a quotient group?

A quotient group is a mathematical structure that is formed by taking a group and "quotienting out" a subgroup. This means that we are grouping together elements of the original group based on their similarity to elements in the subgroup.

2. How is the order of a group defined?

The order of a group is defined as the number of elements in the group. It is also referred to as the size or cardinality of the group. In other words, it is the total number of elements that are present in the group.

3. What does it mean for a group to have finite order?

A group has finite order if it has a finite number of elements. This means that the group has a specific, finite size and is not infinite.

4. How is the order of an element in a group related to the order of the group?

The order of an element in a group is the smallest positive integer n such that the element raised to the nth power is equal to the identity element of the group. The order of the group is the number of elements in the group. These two values can be related through the Lagrange's Theorem, which states that the order of any subgroup of a group must divide the order of the group.

5. What does it mean for an element in a quotient group to have finite order?

If an element in a quotient group has finite order, it means that the element has a finite number of distinct powers. In other words, there is a specific positive integer n such that the element raised to the nth power is equal to the identity element of the quotient group.

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