Proving the Exponent Laws in Group Theory

[Bashyboy]

Homework Statement

Let $x \in G$ and $a,b \in \mathbb{Z}^+$ Prove that $x^{a+b} = x^a x^b$.

Homework Equations

The Attempt at a Solution

If I am not mistaken, we would have to do multiple induction on $a$ and $b$ for the statement/proposition $P(a,b) : x^{a+b} = x^a x^b$. First we would have to show that $P(a,1)$ and $P(1,b)$ are true (base cases). But due to symmetry, it suffices to prove $P(a,1): x^{a+1} = x^a x$. This is where I had trouble; I thought I would have to do induction on the statement on the statement $P(a,1)$.

After trying many things, I decided to consult this https://crazyproject.wordpress.com/2010/01/04/laws-of-exponents-in-a-group/ This person appears to be doing the same as myself. But I found it unsettling that no justification was given for $x^{a+1} = x^a x$ (the base case); what is written is this: "For the base case, note that

for all

."

The only way I could think of justifying it is as follows:

$x^{a+1} = \underbrace{x \cdot x \cdot ... \cdot x}_{(a+1)~terms}$

$= \underbrace{(x \cdot x \cdot ... \cdot x)}_{a ~terms} x$ (associative property)

$= x^a x$ (by definition of what $x^a$ means)But if we can do that, then why not do similar splitting with $a^{a+b}$ , thereby avoiding all induction?

 

[member 587159]

Define G

 

[Bashyboy]

Dummit and Foote doesn't specify, but I imagine they are regarding it as a group.

 

[SammyS]

Homework Statement

Let $x \in G$ and $a,b \in \mathbb{Z}^+$ Prove that $x^{a+b} = x^a x^b$.

Homework Equations

The Attempt at a Solution

If I am not mistaken, we would have to do multiple induction on $a$ and $b$ for the statement/proposition $P(a,b) : x^{a+b} = x^a x^b$. First we would have to show that $P(a,1)$ and $P(1,b)$ are true (base cases). But due to symmetry, it suffices to prove $P(a,1): x^{a+1} = x^a x$. This is where I had trouble; I thought I would have to do induction on the statement on the statement $P(a,1)$.

I don't see the symmetry off hand, unless you have an "extended" associative property. (I suppose that's always implied.)
At any rate, it looks like induction on b is all that's required.

After trying many things, I decided to consult this https://crazyproject.wordpress.com/2010/01/04/laws-of-exponents-in-a-group/ This person appears to be doing the same as myself. But I found it unsettling that no justification was given for $x^{a+1} = x^a x$ (the base case); what is written is this: "For the base case, note that

for all

."

It looks like they just sketched the proof, leaving the reader to fill in some things they thought to be obvious, or at least fairly straight forward.

The only way I could think of justifying it is as follows:

$x^{a+1} = \underbrace{x \cdot x \cdot ... \cdot x}_{(a+1)~terms}$

$= \underbrace{(x \cdot x \cdot ... \cdot x)}_{a ~terms} x$ (associative property)

That shouldn't even require associativeity if normal order of operations is left to right.

$x^{a+1}= \underbrace{\left(\dots\left(\left(\left(x \cdot x \right)\cdot x\right)\cdot x\right) ... \cdot x\right)}_{a ~\text{factors}} \cdot x$​

.

$= x^a x$ (by definition of what $x^a$ means)

But if we can do that, then why not do similar splitting with $a^{a+b}$ , thereby avoiding all induction?

(Surely you mean $\ x^{a+b}\$ in the above line.)

Induction isn't that difficult for this.

Assume that $\ x^{a+n} = x^a x^n\$ for some n ≥ 1. From that show that $\ x^{a+(n+1)} = x^a x^{n+1}\$ follows.

 

[Bashyboy]

That shouldn't even require associativeity if normal order of operations is left to right.
xa+1=(…(((x⋅x)⋅x)⋅x)...⋅x)a factors⋅xxa+1=(…(((x⋅x)⋅x)⋅x)...⋅x)⏟a factors⋅xx^{a+1}= \underbrace{\left(\dots\left(\left(\left(x \cdot x \right)\cdot x\right)\cdot x\right) ... \cdot x\right)}_{a ~\text{factors}} \cdot x

So, are you saying that this is a valid way of establishing $x^{a+1} = x^a x$? If so, then why can't I do the following:

$x^{a+b} = \underbrace{x \cdot x \cdot ... \cdot x}_{(a+b)-terms}$

$= \underbrace{x \cdot x \cdot ... \cdot x}_{a~terms} \cdot \underbrace{x \cdot x \cdot ... \cdot x}_{b~terms}$#

$=x^a x^b$,

thereby avoiding induction.

 

[SammyS]

So, are you saying that this is a valid way of establishing $x^{a+1} = x^a x$? If so, then why can't I do the following:

$x^{a+b} = \underbrace{x \cdot x \cdot ... \cdot x}_{(a+b)-terms}$

$= \underbrace{x \cdot x \cdot ... \cdot x}_{a~terms} \cdot \underbrace{x \cdot x \cdot ... \cdot x}_{b~terms}$#

$=x^a x^b$,

thereby avoiding induction.

I'm not saying you can or you can't do that.

It's partly a matter of what you have previously proved and what you have not proved.

Doing the inductive proof doesn't require you to assume much of anything.

Do you understand inductive proof ?

 

[Bashyboy]

I am still wondering about the justification of the base case $x^{a+1} = x^a x$.

 

[SammyS]

I am still wondering about the justification of the base case $x^{a+1} = x^a x$.

I probably should have restated that it's not the associative law. It's basically just the definition of what is meant by $\ x^n\,,\$ perhaps along with the standard order of operations being left to right.

 

[Bashyboy]

And we agree that the definition of $x^n$ is $x^n = \underbrace{x \cdot x \cdot ... \cdot x}_{n-terms}$? If so, then for $n=a+b$

$x^{a+b} = \underbrace{x \cdot x \cdot x ... \cdot x}_{(a+b)-terms}$

$= \underbrace{x \cdot x \cdot ... \cdot x}_{a} \cdot \underbrace{x \cdot x \cdot ... \cdot x}_{b}$.

$=x^a x^b$.

which is my problem. There is some subtle logic that is preventing me from seeing the need of any induction, as well as how we justify $x^{a+1} = x^a x$.

 

[vela]

The subtlety may be that you have to prove the generalized associative property. You know (ab)c = a(bc), but what about when you have more than three elements?

 

[Bashyboy]

Yes, we do have the generalized associative property; this is proved in the chapter before this problem.

 

[vela]

I just ran across a proof. In that class, $a^n = a^{n-1}a$ held by definition (along with $a^0 = e$ and $a^{-n} = (a^{-1})^n$) for $n \ge 1$.

 

[Bashyboy]

Where did you find this? In my Dummit and Foote book, the only thing that holds by definition is $a^0 = e$.

 

[vela]

In notes for an abstract algebra class. You probably have some way to define what $x^n$ means other than the phrase "multiply $x$ together $n$ times."

 

[vela]

I found a copy of your book online, and it appears it is defined by that phrase. In that case, I think your non-inductive proof is fine.

On the other hand, if $x^n$ is defined as $x^{n-1}x$ with $x^0=e$, then I can see where you'd want to prove the law by induction.