Proof of x^a Inverse in Arbitrary Multiplicative Group

In summary, the conversation discusses the relationship between x and its powers in terms of the arbitrary multiplicative group. The discussion centers around proving that (x^a)^-1 = x^-a using induction and the given equations. However, one of the participants expresses doubt in the validity of the proof due to a lack of justification for a specific step.
  • #1
icantadd
114
0

Homework Statement



x is in the arbitrary multiplicative group, and a,b are positive integers.
given that
[tex] x^{a+b} = x^ax^b[/tex] and [tex] (x^a)^b =x ^{ab} [/tex]
show
that
[tex] (x^a)^-1 = x^{-a} [/tex]

Homework Equations


na



The Attempt at a Solution


Induction:

I) (x)^{-1} = x ^{-1}
II) Assume [tex] (x^n)^{-1} = x^{-n} [/tex], to prove that [tex] (x^{n+1})^{-1} = x^{-(n+1)} [/tex].

[tex] (x^{n+1})^{-1}) = (x^nx)^{-1} = x^{-1}x^{-n} = x^{-n-1} [/tex]

Is the last step justified?
 
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  • #2
icantadd said:
Is the last step justified?
Seeing how you didn't provide a justification, no. :wink:

More seriously, if you cannot see a rigorous reason why that last step should be true, then you definitely haven't written a valid proof.
 
  • #3
Hurkyl said:
Seeing how you didn't provide a justification, no. :wink:

More seriously, if you cannot see a rigorous reason why that last step should be true, then you definitely haven't written a valid proof.

Actually, this is what gets me!

The text explicitly states the following
"[tex] x^{-1}x^{-1}x^{-1} \ldots x^{-1} \text{ n terms }[/tex]"

It should follow from this that
[tex] x^{n} = x^{-1}x^{-(n-1)} [/tex]

I can't see a rigorous road from this description, per se.
 

Related to Proof of x^a Inverse in Arbitrary Multiplicative Group

1. What is "Proof of x^a Inverse in Arbitrary Multiplicative Group"?

"Proof of x^a Inverse in Arbitrary Multiplicative Group" is a mathematical concept that demonstrates the existence and properties of inverse elements in a multiplicative group. It is commonly used in algebra and number theory to show the existence of solutions to equations and to prove theorems.

2. How is "Proof of x^a Inverse in Arbitrary Multiplicative Group" different from other proofs?

"Proof of x^a Inverse in Arbitrary Multiplicative Group" is unique in that it specifically focuses on the inverse elements in a multiplicative group. Other proofs may focus on different properties or operations within a group.

3. What is the importance of understanding "Proof of x^a Inverse in Arbitrary Multiplicative Group"?

Understanding "Proof of x^a Inverse in Arbitrary Multiplicative Group" is important because it allows for a deeper understanding of the structure and properties of multiplicative groups. It also has practical applications in cryptography and coding theory.

4. What are some key steps in the proof of x^a Inverse in Arbitrary Multiplicative Group?

The proof of x^a Inverse in Arbitrary Multiplicative Group typically involves showing the existence of an inverse element, proving its uniqueness, and demonstrating its properties such as the associative and commutative properties.

5. Are there any limitations to "Proof of x^a Inverse in Arbitrary Multiplicative Group"?

Like any mathematical proof, "Proof of x^a Inverse in Arbitrary Multiplicative Group" may have limitations in certain scenarios or for certain types of groups. It is important to carefully consider the assumptions and conditions used in the proof to determine its applicability.

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