Prove Group Commutativity: (G,*) w/ x*x=eG

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In summary, (A) To prove that G is commutative, let (x,y) be in G and use the property (xy)^{-1} = y^{-1}x^{-1} to show that xy = yx. (B) A specific example of an infinite group (G,*) that satisfies this property is the group of integers under multiplication.
  • #1
tasha10
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(A) Let (G,*) be a group such that x*x=eG for all x in G. Prove G is commutative.
(B) Give a specific example of an infinite group (G,*) such that x*x=eG for all x in G.

I have not gotten very far, just to let two variable x,y be in G and I know that (x*y)*(x*y) = eG .. I'm not sure where to go from here..
 
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  • #2
Well, for a, you know ...
[tex]
(xy)^{-1} = y^{-1}x^{-1}
[/tex]

Try multiplying that to both sides of the equality you presented, and see what you get.
 
  • #3
hmm can i ask, what's "eG" means?? identity?
 
  • #4
so, will i just get eG on both sides? does this prove that it is commutative? I'm confused.
 
  • #5
yes, it is the identity
 
  • #6
You won't get eG on both sides. I'm saying, multiply what I showed you to both sides of

[tex]
xyxy = e_{G}
[/tex]
 

1. What does "Prove Group Commutativity" mean in this context?

In mathematics, a group is a set of elements with a binary operation (usually denoted by *) that follows certain rules, such as associativity, identity, and inverse properties. Commutativity refers to the property of the operation where the order of the elements does not affect the result. Therefore, proving group commutativity means showing that for any two elements x and y in the group (G,*), the operation * is commutative, i.e. x*y = y*x.

2. What is G and * in the statement "Prove Group Commutativity: (G,*) w/ x*x=eG"?

G represents the set of elements in the group and * represents the binary operation defined on the elements in G. In this context, we are proving group commutativity for a specific group (G,*) where * is the operation and G is the set of elements.

3. What does x*x=eG mean in the statement "Prove Group Commutativity: (G,*) w/ x*x=eG"?

x*x=eG means that for any element x in the group G, multiplying x by itself using the operation * results in the identity element (eG) of the group. The identity element is the element that, when combined with any other element in the group using the operation *, results in that same element. In other words, multiplying x by itself "cancels out" and returns the original element.

4. Why is it important to prove group commutativity?

Proving group commutativity is important because it allows us to simplify calculations and make predictions about the behavior of the group. If a group is commutative, we can rearrange the order of elements in the operation without changing the result, which can be useful in solving problems in various fields, such as algebra, physics, and computer science.

5. What is the process of proving group commutativity?

The process of proving group commutativity involves showing that for any two elements x and y in the group (G,*), x*y = y*x. This can be done by using the properties of the group, such as associativity and inverse elements, and manipulating the expression x*y = y*x until it matches the definition of commutativity. This process may also involve using specific examples or counterexamples to demonstrate the commutative property for a particular group.

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