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In documents relating to group theory it seems common to use a multiplicative notation to represent the group operation. For example, I'm reading Herstein's "Topics in Algebra" and looking for some pointers about vector spaces in the section on groups. In the vector space section the group operation combining vectors is (quite logically) represented as addition ($v = v_1 + v_2$), but when I switch across to the section on groups it's multiplication ($c = ab$).

Besides finding the switch of notation unhelpful, I feel that the additive notation is a better analogy with real arithmetic: all group elements have an inverse as they do with addition, whereas the multiplicative notation carries an untrue suggestion that there may be a "0" which has no inverse.

So, have I missed something: is there some reason why the multiplicative notation is preferable ?

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    $\begingroup$ Just a guess: This is consistent with the conventions in ring theory: When considering rings, the 'addition' is always commutative, while the multiplication is not. So in group theory you usually do not assume commutativity, but if you use the additive notation, then it is clear that you are talking about a commutative group (and many groups can be considered as a ring but ignoring multiplication). So I think this convention makes perfectly sense. $\endgroup$ – flawr Jan 29 '15 at 14:30
  • $\begingroup$ I agree with @flawr. The additive notation suggests commutativity. If a group is commutative, then $+$ is used very frequently. But when we consider abstract groups $+$ could be misleading. $\endgroup$ – Janko Bracic Jan 29 '15 at 14:33
  • $\begingroup$ My guess is that group theory started with rotations and reflections, with reflections being seen as negative but the reflection of a reflection being seen as positive. $\endgroup$ – Henry Jan 29 '15 at 14:33
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    $\begingroup$ I'm not sure I'd call this "multiplicative" notation - we don't explicitly write out the multiplication sign. I think that this is a case of juxtaposition having two meanings, multiplication (when used for numeric variables) and an arbitrary group operation. $\endgroup$ – Michael Lugo Jan 29 '15 at 14:45
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    $\begingroup$ Before groups were known to be interesting on their own, permutation groups had been known for quite a while. As these are (invertible) functions on a set, the composition is written as $f\circ g$ or even $fg$ leading to a multiplicative notation. $\endgroup$ – j.p. Jan 29 '15 at 17:01
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Generally, multiplicative notation is used for the operation in an arbitrary group, and additive notation is reserved for the operation in Abelian (commutative groups). For some intuition into this, the general linear group $GL_n(\mathbb R)$ is the set of all invertible $n\times n$ matrices, with the operation of matrix multiplication. If $A,B\in GL_n(\mathbb R)$ then we would denote their product $AB$ or $BA$ (note that these are not equal in general!). If we consider $\mathbb R^n$ as a vector space, then it is an Abelian group under addition - so if $x,y\in\mathbb R^n$ then we write the sum as $x+y$ or $y+x$ (and these are indeed equal).

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You surely know that many groups are not commutative. Because of this, an additive notation would bring confusion in non-abelian context.

Think about group of invertible matrices $\mathrm{GL}(\mathbf{K},n)$ over a field $\mathbf{K}$: this is a group under matrix multiplication: wouldn't be rather tricky to think $A+B$ as the result of $A\cdot B$, knowing that it can be very different from $B+A$?

Inverse elements are also another fact. In general, you can prove that for a group $G$ if $x,y\in G$ then $(xy)^{-1} = y^{-1} x^{-1}$. Imagine this in additive notation: $$ -(x+y)=-y - x$$ and that would be DIFFERENT from $$-x-y=-(y+x)$$

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Let $(G,\star)$ be a group with $x\in G$. Suppose you want to write $x \star x \star \cdots \star x$ ($n$ times).

How would you write it with the multiplicative notation? $x^n$.

How would you write it with the additive notation? You can't say $nx$ or $xn$. You have to use an awkward construction like $\sum_{i = 1}^n x$.

Now lets assume we are working over a vector space with scalars in $\mathbb{R}$. How would we write it now? We would just use $nx$. In contexts where there is a reasonable definition to $nx$, usually the additive notation is used.

Add this to the fact that $+$ is used to mean an abelian groups.

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    $\begingroup$ What do you mean "you can't say $nx$"? For an abelian group, this is exactly the notation that is used. This is because abelian groups are best thought of as modules over $\mathbb{Z}$. $\endgroup$ – 6005 Jan 29 '15 at 16:24
  • $\begingroup$ In other words, everything you say earlier on in your answer really just ties back to whether or not the group is abelian. $\endgroup$ – 6005 Jan 29 '15 at 16:25

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