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

In summary, to prove that the elements of order 2 and the identity element e form a subgroup of a commutative group G, one must verify the properties of associativity, unique inverse, identity element, and closure. In particular, for closure, it must be shown that for any elements a and b in the group, their product ab is also in the group. To do this, one must use the fact that a and b have order 2 and evaluate their product using the commutativity of the group.
  • #1
HuaYongLi
16
0

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.
 
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  • #2
Given any subset [tex]H \subset G[/tex], how would you attempt to prove that it is a subgroup of [tex]G[/tex]? What properties of [tex]H[/tex] would you attempt to verify?
 
  • #3
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.
 
  • #4
You need to prove that if a and b are elements of order 2 (i.e. [itex]a^{2} = b^{2} = e[/itex]), then so is [itex]c = a b[/itex]. You need to evaluate [itex]c^{2}[/itex] and use the commutativity of the group.
 
  • #5
Thank You
 

1. What is a subgroup?

A subgroup is a subset of a group that has the same group operation as the larger group and contains the identity element. It is a smaller group within a larger group.

2. How do you prove that a set of elements is a subgroup of a group?

To prove that a set of elements is a subgroup of a group, you need to show that it satisfies the three conditions of a subgroup: closure, associativity, and existence of an identity element. You also need to show that every element in the subset has an inverse in the subset.

3. What is the order of an element in a group?

The order of an element in a group is the smallest positive integer n such that the element raised to the power of n is equal to the identity element. In simpler terms, it is the number of times you need to combine the element with itself to get the identity element.

4. How do you prove that a subgroup only contains elements of order 2 and the identity element?

To prove that a subgroup only contains elements of order 2 and the identity element, you need to show that every element in the subset has an order of 2 or is the identity element. This can be done by raising each element to the power of 2 and showing that it equals the identity element.

5. Can a subgroup have elements of different orders?

No, a subgroup can only have elements of the same order or the identity element. This is because a subgroup must contain the identity element and the inverse of every element in the subset, which both have an order of 1. Therefore, all elements in the subset must have the same order.

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