MHB Does (ab)^n Always Equal a^n*b^n in an Abelian Group?

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evinda
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Hello! (Wave)

Let $G$ a multiplicative abelian group. Show with induction that for each positive integer $n$, $(a \cdot b)^n=a^n \cdot b^n, \forall a,b \in G$. Then,show that the same relation stands for $n \leq 0$.

That's what I have tried:

For $n=1$: $(a \cdot b)^1=a^1 \cdot b^1 \checkmark$

Suppose that the relation stands for $n=k$, $(a \cdot b)^k=a^k \cdot b^k$

We want to show that the relation stand for $n=k+1$.
$$(a \cdot b)^{k+1}=a^{k+1} \cdot b^{k+1} \Rightarrow (a \cdot b)^{k} \cdot (a \cdot b)=a^k \cdot a \cdot b^k \cdot b \Rightarrow a^k \cdot b^k \cdot a \cdot b= a^k \cdot a \cdot b^k \cdot b \Rightarrow b^k \cdot a= a \cdot b^k \Rightarrow a \cdot b^k=a \cdot b^k \checkmark $$

Now we have shown that $(a \cdot b)^n=a^n \cdot b^n, n \in \mathbb{Z}^{+} $

For $n \leq 0$: $n=-k, k\in \mathbb{Z}^{+}$
$$(a \cdot b)^n=a^n \cdot b^n \Rightarrow (a \cdot b)^{-k}=a^{-k} \cdot b^{-k} \Rightarrow ((a \cdot b)^{-1})^{k}=(a^{-1} \cdot b^{-1})^{k} \Rightarrow (b^{-1} \cdot a^{-1})^{k}=(a^{-1} \cdot b^{-1})^{k} \Rightarrow (b^{-1})^k \cdot (a^{-1})^k= (a^{-1})^k \cdot (b^{-1})^k \Rightarrow (a^{-1})^k \cdot (b^{-1})^k=(a^{-1})^k \cdot (b^{-1})^k \checkmark$$

For $n=0$: $(a \cdot b)^0=a^0 \cdot b^0 \Rightarrow 1=1 \checkmark$

Could you tell me if it is right?? :confused:
 
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evinda said:
Hello! (Wave)

Let $G$ a multiplicative abelian group. Show with induction that for each positive integer $n$, $(a \cdot b)^n=a^n \cdot b^n, \forall a,b \in G$. Then,show that the same relation stands for $n \leq 0$.

That's what I have tried:

For $n=1$: $(a \cdot b)^1=a^1 \cdot b^1 \checkmark$

Suppose that the relation stands for $n=k$, $(a \cdot b)^k=a^k \cdot b^k$

We want to show that the relation stand for $n=k+1$.
$$(a \cdot b)^{k+1}=a^{k+1} \cdot b^{k+1} \Rightarrow (a \cdot b)^{k} \cdot (a \cdot b)=a^k \cdot a \cdot b^k \cdot b \Rightarrow a^k \cdot b^k \cdot a \cdot b= a^k \cdot a \cdot b^k \cdot b \Rightarrow b^k \cdot a= a \cdot b^k \Rightarrow a \cdot b^k=a \cdot b^k \checkmark $$

This is confusing. Your first statement, $(ab)^{k+1}=a^{k+1}b^{k+1}$, is what you're trying to prove! You can't start with that. Perhaps you should reverse the direction of your arrows? I think you could - your steps seem reversible to me.

Now we have shown that $(a \cdot b)^n=a^n \cdot b^n, n \in \mathbb{Z}^{+} $

For $n \leq 0$: $n=-k, k\in \mathbb{Z}^{+}$
$$(a \cdot b)^n=a^n \cdot b^n \Rightarrow (a \cdot b)^{-k}=a^{-k} \cdot b^{-k} \Rightarrow ((a \cdot b)^{-1})^{k}=(a^{-1} \cdot b^{-1})^{k} \Rightarrow (b^{-1} \cdot a^{-1})^{k}=(a^{-1} \cdot b^{-1})^{k} \Rightarrow (b^{-1})^k \cdot (a^{-1})^k= (a^{-1})^k \cdot (b^{-1})^k \Rightarrow (a^{-1})^k \cdot (b^{-1})^k=(a^{-1})^k \cdot (b^{-1})^k \checkmark$$

For $n=0$: $(a \cdot b)^0=a^0 \cdot b^0 \Rightarrow 1=1 \checkmark$

Could you tell me if it is right?? :confused:

The way you're writing this is not easy to follow. Put one step on one line. Or just do one long equality.

We have that $(ab)^{-1}=b^{-1}a^{-1}=a^{-1}b^{-1}$, giving the base case. Assume the result is true for $-k$, where $k\in\mathbb{Z}^{+}$. That is, assume $(ab)^{-k}=a^{-k}b^{-k}$. Then
\begin{align*}
(ab)^{-(k+1)}&=((ab)^{k+1})^{-1} \\
&=((ab)^k(ab))^{-1} \\
&=(ab)^{-1}(a^kb^k)^{-1} \\
&=b^{-1}a^{-1}(b^k)^{-1}(a^k)^{-1} \\
&=a^{-1}a^{-k}b^{-1}b^{-k} \\
&=a^{-(k+1)}b^{-(k+1)},
\end{align*}
as required.
 
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