Finding Elements of Order 2 in Z6

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In summary, we are looking at the problem of finding a subgroup of an abelian group G, specifically the set of all elements of G with order 2. We focus on the case of Z6 and determine that the elements of order 2 in this group are 0 and 3.
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POtment
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Homework Statement


Let G be an abelian group. Show that the set of all elements of G of order 2 forms a subgroup of G. Find all elements of order 2 in Z6.

The Attempt at a Solution


The elements of Z6 are 1,4,5. I'm not sure how to find the set of all elements of order 2. Can someone help with that? I think I can prove from there.
 
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  • #2
POtment said:

The Attempt at a Solution


The elements of Z6 are 1,4,5.

Nope. How did you come up with that? And are you talking about the additive group [itex]\mathbb{Z}_6[/itex], or the multiplicative group [itex]\mathbb{Z}_6[/itex]? It does make a difference.
 
  • #3
We are talking about the additive group of Z6. Can you nudge me in the right direction to find the correct elements?
 
  • #4
If it's additive then an element g has order 2 if g+g=0 and g is not zero. It should be pretty easy to find them. But why do you say the elements are 1,4,5?
 
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  • #5
OK. I understand what you are saying and know how to construct the table to find the elements of order 2. However, I still don't understand what the elements of Z6 are. Is it as obvious as 0,1,2,3,4,5?
 
  • #6
It's as obvious as that, yes. So for which ones is g+g=0 mod 6?
 
  • #7
(0,0), (3,3)
 
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  • #8
This one is solved. Thanks for nudging me in the right direction!
 
  • #9
I didn't ask for which pairs is (a,b) is a+b=0. Why would I?? Look up the definition of 'order 2'. Now write it on a blackboard ten times. Now step back and look at it. Now tell me why most of the pairs in your 'guess' aren't interesting.
 
  • #10
POtment said:
(0,0), (3,3)

Better. So the subgroup is made of the elements 0 and 3.
 

What is the definition of an element of order 2 in Z6?

An element of order 2 in Z6 is an element that, when multiplied by itself, results in the identity element (0). In other words, the order of the element is 2, meaning it takes 2 operations to return to the identity element.

How do you find elements of order 2 in Z6?

To find elements of order 2 in Z6, we can use the formula a^2 = 0 (mod 6). This means that we need to find elements that, when squared, have a remainder of 0 when divided by 6. These elements are 2 and 4, since 2^2 = 4^2 = 0 (mod 6).

Why is it important to find elements of order 2 in Z6?

Finding elements of order 2 in Z6 is important because it allows us to identify elements that have a special property in this mathematical structure. These elements can be used to define subgroups and help solve problems related to cyclic groups.

Can there be more than one element of order 2 in Z6?

Yes, there can be more than one element of order 2 in Z6. In fact, there are two elements of order 2 in Z6, namely 2 and 4. This is because in Z6, every number has an inverse, meaning there is always a number that, when multiplied by the element, results in the identity element.

What other mathematical structures can we find elements of order 2 in?

Elements of order 2 can be found in various mathematical structures, such as groups, rings, and fields. In each of these structures, the definition of an element of order 2 may vary slightly, but the concept remains the same - an element that, when operated on itself, results in the identity element.

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