Proving Group Elements & Inverse Have Same Order

  • Thread starter semidevil
  • Start date
In summary, the conversation discusses the concept of order in a group and how it relates to elements and their inverses. It is stated that in a group, each element has a unique inverse and therefore the same number of elements as inverses. The definition of the order of an element in a group is also mentioned, where it is defined as the smallest positive integer n such that g^n = identity. It is then shown how this definition can be used to prove that an element and its inverse have the same order.
  • #1
semidevil
157
2
so I need to show that in any group, it's elements and inverse has the same order.

so can I say that since it is a group, we know that there exists a unique inverse for each element. So each element would have 1 inverse. And then, that means we have the same number of elements as number of inverses?

does that work? or am I missing something?
 
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  • #2
Can you state the definition of the order of an element in a group?
 
  • #3
shmoe said:
Can you state the definition of the order of an element in a group?


so the order is the number of elements in a group...

is this suppose to be a hint? :)
 
  • #4
semidevil...

The order of an element g in a group G is n such that g^n = identity where n is the smallest positive integer >= 1. To prove an element and its inverse have the same order you can say:

g^n = identity
(g^-1) ^ n = (g^n)^-1 = (identity)^-1 = identity
 
  • #5
semidevil said:
so the order is the number of elements in a group...

That's the order of the group, not an element. I was hoping to get you to check the definition carefully;).
 
  • #6
Heh, sorry shmoe...didn't mean to steal your thunder.
 
  • #7
No thunder to be stolen, we're all here to help (or be helped) :smile:
 

1. What is the definition of "order" in group theory?

In group theory, the order of an element is defined as the smallest positive integer n such that the element raised to the power of n equals the identity element.

2. How can we prove that the order of a group element is equal to the order of its inverse?

To prove that the order of a group element is equal to the order of its inverse, we can use the fact that the inverse of an element raised to a power is equal to the inverse of that power of the element. Therefore, if the element has order n, its inverse will also have order n.

3. Can we use induction to prove that the order of a group element is equal to the order of its inverse?

Yes, we can use mathematical induction to prove that the order of a group element is equal to the order of its inverse. This involves proving the base case for n = 1, and then using the inductive hypothesis to prove the statement for n+1.

4. What is the significance of proving that the order of a group element is equal to the order of its inverse?

Proving that the order of a group element is equal to the order of its inverse is significant because it provides a fundamental property of group elements. It also allows us to make conclusions about the structure of groups and their subgroups.

5. Are there any exceptions to the rule that the order of a group element is equal to the order of its inverse?

No, there are no exceptions to this rule. In fact, this is one of the defining properties of a group. If the order of a group element is not equal to the order of its inverse, then the structure does not satisfy the axioms of a group.

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