Are There Common Patterns in Group Elements and Associativity Axioms in Tables?

[dijkarte]

While studying groups,

Is there a common pattern/arrangement of the group elements represented in a table?

Is there a pattern for the associativity axiom?

Thanks.

 

[Number Nine]

A pretty interesting property is that the multiplication table of a (finite) group is a latin square.

 

[dijkarte]

In addition to this, another property I noticed is that there must be one column and one row containing elements in the same order as listed in the table header, and correct me if I'm wrong...

But what's the pattern for associativity?

 

[dodo]

But what's the pattern for associativity?

No idea... I don't think there is one.

But a noticeable pattern in any group table is that no element repeats in any row (or column). A proof of that is a nice, short exercise.

 

[dijkarte]

Cancellation law does it.

r * c = x
r * c[j] = x

==> r * c = r * c[j]
==> c = c[j] (cancellation law)
which is not allowed, repeating elements of the set.

 

[dodo]

In other words, each row (or column) is a permutation of the elements of the group... a fact that will have some protagonism later on in your course, so remember this. (I don't want to spoil the ending of the movie :)

 

[dijkarte]

I see it as a restricted form of permutation where it depends on the selected permutations of other rows/columns.

 

[dijkarte]

How many finite groups can be generated from a set S and an operation * defined on S such that they are of order |s|, same number of elements?

I looked at Dummitt and Fotte quickly but it does not seem to have any illustrations and tables...barely mentioned in exercises which have no solutions. Is there any text that goes deeper into details?

 

[Stephen Tashi]

it does not seem to have any illustrations and tables...barely mentioned in exercises which have no solutions. Is there any text that goes deeper into details?

I don't know of any standard text on group theory that emphasizes the multiplication tables of groups. I think the reason for this is that most of the significant theorems about the structure of groups don't tell you how to deduce their multiplication tables. It also isn't simple to recognize many important properties of a group by looking for a certain pattern in its multiplication table.

An elementary book about groups with some visual appeal is "Groups and Their Graphs" by Grossman and Magnus. However, it doesn't emphasize the multiplication tables of groups.

 

[Number Nine]

This kind of subject is really more the domain of combinatorics than abstract algebra, since most interesting (algebraic) properties of groups can't necessarily be deduced from their Cayley tables. You might try picking up, say, Cameron's Combinatorics, which has some good chapters on latin squares, quasi-groups, and finite geometry that tackle these kinds of things.