Prove that elements of x^i 0<i<n-1 are distinct

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In summary, if x is an element of finite order ##n## in ##G##, then the elements ##1,x,x^2, \dots , x^{n-1}## are all distinct.
  • #1
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Here is the statement: If x is an element of finite order ##n## in ##G##, prove that the elements ##1,x,x^2, \dots , x^{n-1}## are all distinct.

So I started by reinterpreting the question into something more tangible. Is it true that this problem is equivalent to the following? Suppose that ##|x| = n## and that ##0 \le k,m \le n-1##. Prove that ##x^k = x^m ## if and only if ##k = m##.
 
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  • #2
Yes, this is the same. You want to prove:
If ##\operatorname{ord}(x)=n ## then ##x^i \neq x^j##.
and substituted this by:
If ##\operatorname{ord}(x)=n ## then ##(x^i = x^j \Leftrightarrow i=j)##
which is the same. As ##"\Leftarrow"## is trivial, the second statement reduces to:
If ##\operatorname{ord}(x)=n ## then ##(x^i = x^j \Rightarrow i=j)##
which is the same as the first statement.

You can either prove the last statement, or assume ##x^i=x^j## for ##i \neq j## and deduce a contradiction to the minimality of ##n##.
 
  • #3
fresh_42 said:
Yes, this is the same. You want to prove:
If ##\operatorname{ord}(x)=n ## then ##x^i \neq x^j##.
and substituted this by:
If ##\operatorname{ord}(x)=n ## then ##(x^i = x^j \Leftrightarrow i=j)##
which is the same. As ##"\Leftarrow"## is trivial, the second statement reduces to:
If ##\operatorname{ord}(x)=n ## then ##(x^i = x^j \Rightarrow i=j)##
which is the same as the first statement.

You can either prove the last statement, or assume ##x^i=x^j## for ##i \neq j## and deduce a contradiction to the minimality of ##n##.
So suppose that ##1 \le i,j < n## and that ##x^i = x^j##. Then ##x^{i-j}=1##. So ##n~ |~ (i-j)## which means that ##i \equiv j \pmod n##. But since ##1 \le i,j < n## we have that ##i=j##. Does this work?

My only problem is the statement that ##i \equiv j \pmod n## and ##1 \le i,j < n## implies ##i=j##. This implication seems kind of obvious to me, but would there be a more formal way of doing it? I haven't taken much number theory so I'm not sure.
 
  • #4
Mr Davis 97 said:
So suppose that ##1 \le i,j < n## and that ##x^i = x^j##. Then ##x^{i-j}=1##. So ##n~ |~ (i-j)## which means that ##i \equiv j \pmod n##. But since ##1 \le i,j < n## we have that ##i=j##. Does this work?

My only problem is the statement that ##i \equiv j \pmod n## and ##1 \le i,j < n## implies ##i=j##. This implication seems kind of obvious to me, but would there be a more formal way of doing it? I haven't taken much number theory so I'm not sure.
No, that's fine. You could do it per pedes (by foot) if you like: Assume ## i > j## then ##0 \neq i-j = qn \geq n ~\Longrightarrow ~i \geq j+n \geq 0+n = n## contradicting ##i<n##. However, this isn't necessary. Such situations are what calculations with modulo are made for.
 
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1. What does "elements of x^i" mean?

The notation x^i represents the i-th power of x. So, elements of x^i refer to the individual values obtained by raising x to different powers, i.e. x^0, x^1, x^2, etc.

2. What does "0

Here, i represents the index of the element, and n is the total number of elements. So, 0

3. How can we prove that the elements of x^i are distinct?

We can prove this by contradiction. Assume that there exist two elements x^i and x^j, where i≠j, but x^i = x^j. This would mean that x^(i-j) = 1, which is only possible if i-j=0, but that contradicts our initial assumption that i≠j. Therefore, the elements of x^i must be distinct.

4. Can you provide an example to illustrate this concept?

Sure, let's take the example of x={2, 4, 8}. Here, n=3. So, from the given statement, we know that the elements will be distinct if i=0, 1, 2. We can easily see that x^0=1, x^1=2, and x^2=4, which are all distinct values.

5. How does this concept relate to mathematics or science?

This concept is important in various mathematical and scientific fields, such as algebra, number theory, and physics. It helps in understanding and solving various problems that involve powers and exponents, and also plays a crucial role in the study of patterns and sequences.

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