Group Theory: Elements as Ops, Representations as Math?

[kent davidge]

Given a group, can we regard its elements as statements of what the operation does, while its representations are the mathematical translation?

For instance, given a square, I'd say that $a$ is an operation that rotates the square by 90°, and the representation of $a$ would be the matrix that actually rotates its vertices as long as coordinates are given.

 

[fresh_42]

Given a group, can we regard its elements as statements of what the operation does, while its representations are the mathematical translation?

For instance, given a square, I'd say that $a$ is an operation that rotates the square by 90°, and the representation of $a$ would be the matrix that actually rotates its vertices as long as coordinates are given.

Given your example, well, yes. Given your question, not really. A group itself is nothing but a set, a binary operation on this set, which additionally obeys some rules. But even given the elements were all rotations, you still have to describe the binary operation, usually as successive application of these operations, but this isn't automatically the case. And if it is the case, we still will have to state what to rotate where and how. Then, and only then, we can check the rules. To write them as matrices is one way to do all this: within a given coordinate system.

Formally a representation of a group $G$ is a group homomorphism $G \stackrel{\varphi}{\longrightarrow} \operatorname{Aut}(X)$ into the group of automorphisms on a set $X$, often regular linear transformations $GL(V)$ of a vector space $X=V$, the representation space. An equivalent wording is to say that $G$ operates on $X$, by $g.x := \varphi(g)(x)$. It is the exact same thing. Now whether we write the elements of $\operatorname{Aut}(X)$ as matrices or otherwise, is a separate question. And of course the group itself doesn't need to be defined by using some specific representation. Sometimes it is natural to do so as in your example with the rotations, but for many others this is not the case, e.g. the integers or polynomials. Your example works, because the group homomorphism $\varphi$ is an embedding $G \subseteq \operatorname{Aut}(\square)$ or $G \subseteq GL(\mathbb{R}^2)$ if you consider the square as part of the plane and you rotate the entire plane, which is a vector space.

 

[Mark44]

Given a group, can we regard its elements as statements of what the operation does, while its representations are the mathematical translation?

I'm not sure this is the best way to think about a group. As already mentioned, a group is a set of things, together with an operation. A given group might represent several situations, so I don't understand your connection of representations with mathematical translations.

For instance, given a square, I'd say that $a$ is an operation that rotates the square by 90°, and the representation of $a$ would be the matrix that actually rotates its vertices as long as coordinates are given.

You example could be represented as $G = \{e, \sigma, \sigma^2, \sigma^3; \cdot\}$. Here e is the identity rotation, $\sigma$ is a rotation by 90°, $\sigma^2$ is rotation by 180°, and so on. The group operation is multiplication, indicated by $\cdot$.
A rotation could be represented by a permutation matrix $\sigma$, where $\sigma = \begin{pmatrix} 1 & 2 & 3 & 4 \\ 2 & 3 & 4 & 1 \end{pmatrix}$. The numbers represent the four vertices, numbered in order, clockwise or counterclockwise -- your choice. If we do one rotation, vertex 1 goes to where vertex 2 was, vertex goes to where vertex 3 was, vertex 3 goes to where vertex 4 was, and vertex goes to where vertex 1 was.

Here's a link to the wiki page on permutation groups: https://en.wikipedia.org/wiki/Permutation_group

 

[mathwonk]

Although as fresh says the formal definition of a group is in terms of certain axioms, in reality they arise as you suggest as actions on some set, such as rotations in space, or permutations of a set. the word "representation", as fresh also says, usually means "linear representation", which is a model of the group as linear actions on vector space, which can be given as matrices in terms of a basis. so yes in some intuitive sense, groups do arise as actions on some set, of any kind, and their numerical representation is usually in terms of linear actions on some vector space. but since rotations of space are in fact linear, in the abstract sense, your group of rotations is already given as a linear representation. i.e. not all linear actions on vector spaces are represented via matrices, just those on vector spaces that have a given basis.

 

[fresh_42]

Although as fresh says the formal definition of a group is in terms of certain axioms, in reality they arise as you suggest as actions on some set, such as rotations in space, or permutations of a set. the word "representation", as fresh also says, usually means "linear representation", which is a model of the group as linear actions on vector space, which can be given as matrices in terms of a basis. so yes in some intuitive sense, groups do arise as actions on some set, of any kind, and their numerical representation is usually in terms of linear actions on some vector space. but since rotations of space are in fact linear, in the abstract sense, your group of rotations is already given as a linear representation. i.e. not all linear actions on vector spaces are represented via matrices, just those on vector spaces that have a given basis.

Thanks for the hint. I should have written $\operatorname{Aut}$, which I corrected now.

 

[Stephen Tashi]

For instance, given a square, I'd say that $a$ is an operation that rotates the square by 90°, and the representation of $a$ would be the matrix that actually rotates its vertices as long as coordinates are given.

If you are studying groups in the context of chemistry ("point groups") you will encounter the confusion between a "group" and a "group action". In many contexts, what writers call a "group" is, mathematically speaking, a "group action".

As other posters indicated , the definition of a "group" in the mathematical sense does not specify its operation to have a specific interpretation as a motion or movement. A common application of groups is to consider situations where their abstract operation does have such an interpretation. Those applications involve "group actions".

There is a theorem called Cayley's theorem that says the operation of a group can be implemented in a specific way by defining each element of the group as a function of the group onto itself. (e.g. The element #a# is identified with the function $A(x) = (a)(x)$ where $(a)(x)$ denotes the result of the abstract group operation. ) We implement the abstract operation of the group by the composition of functions. (e.g. $(A)(B)$ denotes the function $A(B(x))$). Insofar as one may consider a function $f$ to implement some sort of "motion" from $y$ to $f(y)$, this shows that the operation of any group can be implemented as that type of motion.

Implementing the operation of a group by a specific set of functions or "motions" is a "group action". (You can look up the precise definition of "group action" and compare it to the definition of a group.)

The technical definition of a "representation" of a group says it is a very particular type of "group action". (To emphasize: a "group representation" is not a "group". It is a type of "group action") A "representation" must be an implementation of the group operation using functions that are linear transformations on a vector space. A given linear transformation on a finite dimensional vector space can be implemented by the operation of multiplying a matrix times a vector.

Consider the situation where we have a group whose elements are denoted by letters "a", "b", ... We don't necessarily have any information that tells us what the notation "5a + 3b" would mean. The elements of an abstract group need not be vectors in some vector space. If someone hands us the multiplication table for a group and says "Here, find a set of matrices whose multiplication implements this table", the problem isn't trivial. By contrast if he says "Find a set of functions that implement this table", he gives us freedom about the domain and co-domain of the functions. We could use Caley's theorem to implement the table as a set of functions that map the group onto itself.

 

[kent davidge]

Thank you guys.
All of these is useful information for me.

 

[mathwonk]

i want to recall that in historical terms, a group actually originated as an action on a set, in the context of an action on roots of an equation, in galois's and others' work on solving equations. then later someone codified a group as a set with an operation (unspecified) satisfying certain axioms. so a group started out as a group action and then later was abstracted into a modern "group" and then again recast as a group action.