Groups - multiplicative notation in additive groups Is it technically incorrect to use multiplicative notation when we are referring to an additive group? For example, consider this group: $$G =(\{0,1,\dots,n-1\}, +_n)$$ where $+_n(x,y) =(x+y)
\mod n$
Assume that $a \in G$ 
Let's perform $a +_na $ k times.
 
In additive notation our result is $ka$ whereas in multiplicative notation it's $a^k $ 
Can we therefore assume that either notation is correct and say that in $G$
$$ka=a^k$$
?
 A: There's nothing to stop you from defining $ab$ to be the same thing as $a+b$ but I can't think of any case where it would be useful to do so.  You're thinking of one operation, there's no reason to use two notations for it simultaneously.
Typically multiplicative notation is used for Abelian groups when you don't yet know they are Abelian, e.g. you are looking at a presentation and trying to figure out what group it presents.  See this MSE question for a group that's not obviously additive/Abelian.
Using the "$+$" and "$ka$" is preferred for additive groups, and suggests thinking of them as $\mathbb{Z}$-modules.
A: Technically, you can use whatever notation you like, as long as you make clear what it means. You can even use different notations for the same thing.
In your specific case though it would usually be quite unfortunate to use the notation you propose, since in the example you give 'multiplication' already has a standard meaning, namely multiplication modulo $n$.  Thus, seeing $3^2$ everybody will assume it is $9$ and not $6$. 
To make this more clear, the very same question could be asked for the real numbers or the integers. You could say that you denote the sum of $a$ and $b$ by $a\div b$. And then $4+2 = 4 \div 2$. What you must not forget though is that then the expression $4 \div 2$ has not its usual meaning anymore. It is just another way to write $4+2$. 
Let me add though that there are case where it makes sense to replace notations. For example in tropical geometry one often denotes the sum of two real numbers by $a \otimes b$, as 'sum of real numbers' is the multiplication in the semiring one considers the additive operation is given by $a\oplus b=\max(a,b)$.    
