Exercise 241Algebra › GroupsHigh schoolAdded 3 Nov 2026

Additive and multiplicative notations

#notation#abelian group#convention

Statement

Let $G$ be a group. Rewrite each statement in multiplicative notation as its additive counterpart (assuming $G$ is abelian):

$(ab)^n=a^nb^n$ ;
$a^{-1}b^{-1}=(ab)^{-1}$ ;
$a^m a^n=a^{m+n}$ ;
the identity is $e$ .

Hint

Replace product by sum, $a^n$ by $na$ , $a^{-1}$ by $-a$ , and $e$ by $0$ .

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Solution

1. $n(a+b)=na+nb$ .

2. $(-a)+(-b)=-(a+b)$ .

3. $ma+na=(m+n)a$ .

4. The identity is $0$ .

Conclusion. The translation between multiplicative and additive notation is purely cosmetic. In additive notation, the group operation is written $+$ , the identity $0$ , and the inverse $-a$ .

Additive and multiplicative notations

Statement

Solution

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