# Abelian group

## 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.

## Answers and Replies

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Tom Mattson
Staff Emeritus
Gold Member

## 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 $\mathbb{Z}_6$, or the multiplicative group $\mathbb{Z}_6$? It does make a difference.

We are talking about the additive group of Z6. Can you nudge me in the right direction to find the correct elements?

Dick
Homework Helper
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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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?

Dick
Homework Helper
It's as obvious as that, yes. So for which ones is g+g=0 mod 6?

(0,0), (3,3)

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This one is solved. Thanks for nudging me in the right direction!

Dick