How Does Group Theory Explain the Equation (a^(-1)*ba)^n = a^(-1)*b^n*a?

[Benzoate]

Homework Statement

For any elements a and b from a group and any integer n, prove that (a^(-1)*ba)^n = a^(-1) b^n *a

Homework Equations

There are no equations given for this particular problem

The Attempt at a Solution

by law of exponents and the distributive property(a^-1 *b*a)= a^-n *b^n *a^n=a^(-n +n)*b^n = a^0 * b^n = b^n

Likewise, a^(-1) * b^n * a^1 = a^(-1+1)* b^n

since (a^-1 *b *a)^n =b^n and since a^-1 * b^n *a^1 = b^n , then (a^-1 * b*a) = (a^-1 *b^n *a^1)

 

[matt grime]

Groups are not commutative. You can't just say, as you have, that (xy)^n=x^ny^n, that is just not true. So what is the definition of x^n? It is x*x*x*..*x with n x's. End of story.

 

[dextercioby]

$$\left(a^{-1}ba\right)^{n}= \left(a^{-1}ba\right) \left(a^{-1}ba\right) \left(a^{-1}ba\right)...$$

Now use associativity.

 

[Benzoate]

$$\left(a^{-1}ba\right)^{n}= \left(a^{-1}ba\right) \left(a^{-1}ba\right) \left(a^{-1}ba\right)...$$

Now use associativity.

Assiociative property: ((a^-1 *b)*a)^n= (a^-1 *(b*a))^n : that would be how you would apply the assiociative property to this problem.

I know the assiociative property is one of the 3 properties, along with the identity and inverse property to prove that a particular non-empty set is a group.

But how is the assiociative property related to what you wrote : ((a^-1 *b)*a)^n=(a^-1 *b)*a)*(a^-1 *b)*a)*(a^-1 *b)*a)

 

[matt grime]

Associativity means that one does not need brackets.

 

[Benzoate]

Associativity means that one does not need brackets.

What do you mean associativity does not need brackets?

Assiociativity is based on this fundamental operation: (ab)c=a(bc) for all a,b, c in G

 

[d_leet]

What do you mean associativity does not need brackets?

Assiociativity is based on this fundamental operation: (ab)c=a(bc) for all a,b, c in G

Yes, and there is no doubt at all that matt knows this. Let's take it one step at a time. Can you calculate/simplify (a^-1ba)²=(a^-1ba)(a^-1ba)?

Hint: apply the associative property.

 

[Benzoate]

Yes, and there is no doubt at all that matt knows this. Let's take it one step at a time. Can you calculate/simplify (a^-1ba)²=(a^-1ba)(a^-1ba)?

Hint: apply the associative property.

yes.

 

[d_leet]

yes.

Ok... And what is it?

 

[Benzoate]

Ok... And what is it?

Why do I need to simplify (a-1ba)^2=(a-1ba)^1 *(a-1ba)^1 again when you already simplfied the equation for me? I thought you wanted me to regconize whether the particular equation you wrote was simplified or not.

 

[d_leet]

Why do I need to simplify (a-1ba)^2=(a-1ba)^1 *(a-1ba)^1 again when you already simplfied the equation for me?

I don't recall doing this, I asked you if you could simplify it yourself and you replied yes, so what did you come up with?

The point of that rather simple question was that if you cannot do that, there is little hope of solving the more general problem in your first post.

So shall we try this again if we have (a^-1ba)²=(a^-1ba)(a^-1ba)

Can you simplify this? Your first post says that (a^-1ba)ⁿ=(a^-1bⁿa) for all natural n. So in other words (a^-1ba)² should equal (a^-1b²a).

Now can you show this?

 

[matt grime]

What do you mean associativity does not need brackets?

Assiociativity is based on this fundamental operation: (ab)c=a(bc) for all a,b, c in G

That is not an operation. Now, it is an identity, which means we can unambiguously write abc, and we know it has a unique meaning.

Perhaps you would prefer it, and find it helpful, if I pointed out that associatvity means we can insert and move brackets around at will in an expression. Now, please, simplify

$$aba^{-1}aba^{-1}$$

Notice how I'm omitting the brackets?