Why Denote Group Operation with Multiplication?

In summary, the use of multiplication notation to denote group operation can cause confusion, especially when first learning abstract algebra. This is because most common examples of groups, such as integers and rationals, use addition as their operation. The use of multiplication notation may stem from its connection to rings and linear algebra. However, for practical applications, such as in matrix groups and representation theory, multiplication notation is more intuitive and convenient. It may be beneficial to introduce a new notation, such as ##a\star b##, to avoid confusion and better explain both multiplicative and additive notation in the context of groups.
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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.
 
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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.
 
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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.
 
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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.
 
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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...

Fredrik said:
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.
 
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Is this discussion whether to use [itex] \cdot [/itex] or + to denote the binary operation in a group ?? The only reasonable way is to choose none of the 2, an example would be [itex] \star [/itex], or even better [itex] \ast [/itex].
 
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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)?
 
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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##.
 

1. Why is multiplication used to denote group operation?

The use of multiplication to denote group operation is a convention that has been established in mathematics. It is a convenient and compact way to represent the combination of two elements in a group. Additionally, multiplication is a familiar and widely understood operation, making it easier to understand and work with in group theory.

2. Can other symbols or operations be used to denote group operation?

Yes, other symbols or operations can be used to denote group operation. Some examples include addition, composition, and exponentiation. However, multiplication has become the standard due to its simplicity and effectiveness in representing group operations.

3. What are the advantages of using multiplication to denote group operation?

Using multiplication to denote group operation allows for the use of familiar properties and operations, such as commutativity and associativity, which are essential in group theory. It also allows for a more compact representation of group operations, making it easier to work with complex groups.

4. Is there a specific reason why multiplication is used instead of other operations?

The choice to use multiplication to denote group operation is largely due to historical and cultural reasons. The use of multiplication in mathematics dates back to ancient civilizations and has been used in various mathematical concepts and operations. This familiarity and prevalence of multiplication make it a suitable choice for denoting group operations.

5. Are there any disadvantages to using multiplication to denote group operation?

One potential disadvantage of using multiplication to denote group operation is that it may lead to confusion or misunderstandings, especially for those who are not familiar with group theory. Additionally, some groups may not follow the same properties as multiplication, which can make it more challenging to work with them using this operation.

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