Why Denote Group Operation with Multiplication?

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Why Denote Group Operation with Multiplication??

When groups are introduced in most abstract algebra texts, the operation is denoted by multiplication or juxtaposition and addition notation is reserved for abelian groups.

This seems to cause a lot of unnecessary confusion. Professors often rely completely on modular arithmetic (additive) to motivate arbitrary quotient groups (multiplicative), so students must constantly translate between the two notations. Other key examples of groups are the integers, rationals, reals and complex numbers, all under addition.

When rings are introduced there is even more confusion, since the group aspect of a ring is denoted by addition, and a ring under multiplication is not a group at all.

It seems to me that it would make more sense to introduce monoids with the multiplication/juxtaposition notation (to emphasize connection to rings) and use addition for group notation.

One might argue that it is important to distinguish between abelian and nonabelian groups. In this case, a more acceptable notation for nonabelian groups could be the composition notation of functions. In particular this would highlight the connection to the group of bijections of a set and avoid confusion with multiplication in a ring.

What do you guys think? I'm not too far into math myself, so I might have overlooked a key reason that multiplicative notation is used for groups.

 

[micromass]

It's just notation. I admit it is confusing at first, but you grow used to it.

Maybe you're right that other notations are better, but the notation we use now is standard. It's impossible to change.

 

[Fredrik]

I don't find the different notations for group operations particularly confusing.

Most of the useful groups are matrix groups. It would be very confusing to use the + notation for them, since the group operation is matrix multiplication.

 

[eotwdave]

I suspect the preference for multiplication may be because of linear algebra and matrix multiplication.

I've been getting into linear algebra recently after neglecting it for years and I've been really floored by just how useful matrix multiplication notation is once you pay attention.
Matrix multiplication (among other things) gives a general way to represent the groups of vector rotations and reflections in any dimension.

Speaking of rotations, the group of rotations of the complex roots of unity is one of the jewels of mathematics. Seeing the deep beauty of e^(i*pi) requires understanding that the complex n-th roots of 1 are groups under complex multiplication, which may be the other big reason for prefering multiplication to addition.

 

[pwsnafu]

This seems to cause a lot of unnecessary confusion. Professors often rely completely on modular arithmetic (additive) to motivate arbitrary quotient groups (multiplicative), so students must constantly translate between the two notations. Other key examples of groups are the integers, rationals, reals and complex numbers, all under addition.

While this maybe true in the lecture theatre, in real world applications of group theory (whether in industry or journal papers) you aren't studying (ℝ,+). You list examples that first years are familiar with. But you learn the theory because you want to study other groups. Which leads to...

Most of the useful groups are matrix groups. It would be very confusing to use the + notation for them, since the group operation is matrix multiplication.

Adding to this, if you are doing groups theory, you'll be using representation theory, and the representation is always matrix multiplication.

 

[dextercioby]

Is this discussion whether to use $\cdot$ or + to denote the binary operation in a group ?? The only reasonable way is to choose none of the 2, an example would be $\star$, or even better $\ast$.

 

[jbunniii]

Multiplicative notation has the advantage that one doesn't have to explicitly write the symbol for the group operation. Who wants to have to write things like $a\cdot b \cdot a^{-1} \cdot b^{-1}$ or $a * b * a^{-1} * b^{-1}$ when $aba^{-1}b^{-1}$ conveys the meaning just as effectively (arguably more so because it's easier to read)?

 

[Fredrik]

The approach that I would recommend is to use a new notation like $a\star b$ when we define the term "group", and then immediately explain "multiplicative notation" $ab$ and "additive notation" $a+b$.