Subgroup Orders

1. Sep 29, 2011

pandapatrol26

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

Find all elements x in Z50 such that <x> = <5>

2. Relevant equations none really

3. The attempt at a solution

I thought <5> would be equal to {0, 5, 10, 15 ... 45} but that doesn't seem to be correct... can anyone tell me what I'm doing wrong?

2. Sep 29, 2011

micromass

Staff Emeritus
Why doesn't it seem correct to you?? What you wrote is ok!

3. Sep 29, 2011

pandapatrol26

because i entered the answer in webwork and it says not correct :'( i'm so confuzzled.

4. Sep 29, 2011

I like Serena

Which answer did you enter?

Do note that 5 is not the only x that has <x> equal to the specified set.

5. Sep 29, 2011

pandapatrol26

I entered all the integers in the set like this: 0, 5, 10 etc. to 45.

i added a picture below. imagine the number going up to 45.

6. Sep 29, 2011

I like Serena

All right.
Let's consider x=10.
What is <10>?
Is it the same as {0, 5, 10, 15 ... 45}?

7. Sep 29, 2011

pandapatrol26

i'd say no to that.

but i don't really know how to tell which elements are reduplicates of each other.
just that <10> would be {0, 10, 20 etc.}

8. Sep 29, 2011

I like Serena

Exactly!
So 10 is not a possible solution.
Actually <10>={0,10,20,30,40} to be precise (this is modulo 50).

Any x that might have the proper <x> would have to be one of the elements of {0, 5, 10, 15 ... 45}.
So the question becomes: which of these elements generates the same set?
Obviously 5 will do the trick, but 10 won't.
What do the other elements do?

9. Sep 29, 2011

pandapatrol26

hmm.. 0?

10. Sep 29, 2011

I like Serena

Huh?

<0> = {0}, so it won't do.

11. Sep 29, 2011

pandapatrol26

:/ then i don't really see any other element that is remotely similar except maybe 15?

12. Sep 29, 2011

I like Serena

So what is <15>?
And while we're at it, what is <45>?

Do you know what <x> means? How you calculate the corresponding set?

13. Sep 29, 2011

pandapatrol26

<15> = {0, 15, 30, 45}
<45> = {0, 45}
??

14. Sep 29, 2011

pandapatrol26

<x> is suppose to mean the order of the element, so <5> means any elements with order 5? so.. in terms of group addition, that would mean if x = 5, then it gets added five times before it becomes the identity?
sorry lol i'm so bad at this

15. Sep 29, 2011

I like Serena

Let's see.
<x> = { 0 mod 50, x mod 50, x+x mod 50, x+x+x mod 50, .... }

So <45> = {0, 45 mod 50, 90 mod 50, ... } = {0, 45, 40, 35, 30, 25, 20, 15, 10, 5}
Note that 45 is also the same as -5 modulo 50.

Can you retry <15>?

16. Sep 29, 2011

I like Serena

Errr... no.

#x is the order of element x.
The order of 5 is 10, since 5+5+5+5+5+5+5+5+5+5=0 mod 50.
This is also the size of the group generated by 5.

<x> is the group generated by x, which is {0, x, x+x, x+x+x, x+x+x+x, ....}

I'm afraid you need to get your definitions straight, before you can solve problems using them.

17. Sep 29, 2011

pandapatrol26

i'm kind of confused how u got 45 to be -5 modulo 50?

18. Sep 29, 2011

pandapatrol26

yes i was saying that <5> means an order of 10 since there's {0, 5, 10, ..45} 10 elements in the set before it reaches the identity- which is 50. so at 50 it would go back to 5. but i'm just not really suer how u determine which elements are duplications of one another like how 45 and 15 is the same as 5.

19. Sep 29, 2011

I like Serena

Well, when we're talking about the element 45 of Z50, we're actually talking about the set of all numbers that are equal to 45 mod 50.
Usually this set is actually denoted as $\bar {45}$, meaning {..., -5, 45, 95, ...}, or $\{ 45 + 50q | q \in \mathbb Z\}$.

We've been leaving off the overbar, since that is a bit hard to write down.
But -5 is the same element as 45 (mod 50), since -5 + 50 = 45.

20. Sep 29, 2011

I like Serena

Errr... 45 is not a duplication of 5, since the difference is not a multiple of 50.
55 is a duplication of 5.

(Actually, 45 is the inverse of 5, since their sum is 0 mod 50.)