Proof on Order of Elements in a Group

In summary, the order of a composition of more than one element is the same as the order of the individual elements.
  • #1
jeffreydk
135
0
I'm trying to figure out how to prove the following...

If [tex]a, b \in G [/tex] where G is a group, then the order of [tex]bab^{-1}[/tex] equals the order of [tex]a[/tex].

I'm rather stumped because the group is not necessarily abelian and it seems like it would have to be in order to directly show that you can rearrange b and b's inverse to get rid of them. I'm confused party because I'm not sure if those properties still hold when you're working with the order of the elements. Any help is greatly appreciated, thanks.
 
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  • #2
What is (bab-1)n? And given b, and k in a group G when is it true that bkb-1 is equal to the identity?
 
  • #3
As a hint, note that [tex] (bab^{-1})^2 = (bab^{-1})(bab) = ba(b^{-1}b)ab^{-1} = baeab^{-1} = baab^{-1} = ba^2b^{-1}[/tex], where [tex]e[/tex] is the identity in [tex]G[/tex].
 
  • #4
Oh ok, it just hit me it makes perfect sense. I wasn't thinking hard enough about the actual definition of order. Thank you both for your help.
 
  • #5
let f:G-->G be an automorphism of G. if x has order n, prove f(x) also has order n.
 
  • #6
But in order to demonstrate that b . b^-1 is an automorphism, you would be basically doing the very proof shown above though?
 
  • #7
doing, and understanding WHAT you are doing, are two different things.
 
  • #8
I can show
[tex](bab^{-1})^n = ba^nb^{-1}[/tex]

But how does this show the orders are equal?

Any help would be great please!
 
  • #9
If n is the order of a, what is [itex]ba^nb^{-1}[/itex] ?
 
  • #10
matt grime said:
If n is the order of a, what is [itex]ba^nb^{-1}[/itex] ?

are you asking what the order of [itex]ba^nb^{-1}[/itex] is?

Woudl I be right in saying that it's n, because no matter what n is the b and b^-1 stay the same?
 
  • #11
Is there some sort of theorem for the order of a composition of more than one element?
 
  • #12
Firepanda said:
are you asking what the order of [itex]ba^nb^{-1}[/itex] is?


No, I'm asking you what it is. Let me try to make it even more clear: if n is the order of a, what is a^n? Now, what is ba^nb^-1?
 
  • #13
matt grime said:
No, I'm asking you what it is. Let me try to make it even more clear: if n is the order of a, what is a^n? Now, what is ba^nb^-1?

Ah i gotcha! Kk it all made sense now.

Would you be able to help me here also? :

https://www.physicsforums.com/showthread.php?t=181745

is saying [tex](ab)^{x} = 1.[/tex] the same as saying [tex](ab)^{x} = e[/tex]?

:)
 

1. What is the order of elements in a group?

The order of elements in a group is the number of elements in the group. It is denoted by |G| and is also referred to as the cardinality of the group.

2. How do you prove the order of elements in a group?

To prove the order of elements in a group, you can use the Lagrange's theorem which states that the order of any subgroup of a finite group divides the order of the group itself. You can also use other methods such as direct counting or the properties of cyclic groups.

3. Can the order of elements in a group be infinite?

No, the order of elements in a group cannot be infinite. A group is defined as a set with a binary operation that satisfies certain properties. If the order of elements is infinite, it would violate the closure property of the group.

4. Does the order of elements in a group affect its structure?

Yes, the order of elements in a group can affect its structure. For example, if a group has a prime number as its order, it is a cyclic group and has a simpler structure compared to groups with composite orders.

5. How does the order of elements in a group relate to its subgroups?

The order of elements in a group is closely related to its subgroups. As mentioned earlier, the Lagrange's theorem states that the order of any subgroup divides the order of the group. This means that the order of elements in a subgroup must also divide the order of elements in the group.

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