Group Theory: Proving Subgroup of Elements of Order 2 and e

[HuaYongLi]

Homework Statement

G is a commutative group, prove that the elements of order 2 and the identity element e form a subgroup.

Homework Equations

The Attempt at a Solution

I don't know where to even begin.

 

[ystael]

Given any subset $$H \subset G$$, how would you attempt to prove that it is a subgroup of $$G$$? What properties of $$H$$ would you attempt to verify?

 

[HuaYongLi]

Well I guess associativity, unique inverse and identity element are all trivial.
What I'm having trouble with is closure, proving that for any elements a and b in the group, ab is also in the group.

 

[Dickfore]

You need to prove that if a and b are elements of order 2 (i.e. $a^{2} = b^{2} = e$), then so is $c = a b$. You need to evaluate $c^{2}$ and use the commutativity of the group.

 

[HuaYongLi]

Thank You