Simple explanation of group generators

[resurgance2001]

Hi

I am trying to get a simple grasp of the concept of a group generator and group representations.

As ever wherever I look, I get very mathematical speak definitions such as:

"A representation is a mapping that takes elements g in G into linear operators F that preserve the composition rule". I get the second half of the sentence but I still get stuck conceptually as soon as I read the words: 'a mapping that takes a group of elements into linear operators'

And similarly, I am stuck on the word 'generators.'

Thanks

Peter

 

[MisterX]

If $W$ is the mapping then the composition rule required for a representation is
$$W(g) \cdot W(h) = W(gh) $$
with
$$W: G\to F$$
$$g, h\in G $$

Here, $W(g)$ is the linear operator that corresponds to the element $g$ from $G$.

A representation is a group homomorphism into some linear operator group.

 

[resurgance2001]

A representation can be a set a matrices, yes? Where each matrix represents an element in the group, so that the operation of the group can be reduced to simple matrix multiplication.

 

[resurgance2001]

If $W$ is the mapping then the composition rule required for a representation is
$$W(g) \cdot W(h) = W(gh) $$
with
$$W: G\to F$$
$$g, h\in G $$

Here, $W(g)$ is the linear operator that corresponds to the element $g$ from $G$.

A representation is a group homomorphism into some linear operator group.

"I don't understand this phrase yet: "A group homomorphism into some linear operator group". Can you explain it please with a simple example? Thanks

 

[MisterX]

"I don't understand this phrase yet: "A group homomorphism into some linear operator group". Can you explain it please with a simple example? Thanks

A homomorphism follows the rules for W as above only instead of mapping into a linear operator group, it could be some arbitrary other group.

 

[resurgance2001]

Thanks - so structure of the group is basically conserved.

But then what are group generators? Are they the most basic elements of the group that are needed to generate all the other elements in an infinite group? Do you have generators in a finite group? Thanks

 

[coquelicot]

Thanks - so structure of the group is basically conserved.

But then what are group generators? Are they the most basic elements of the group that are needed to generate all the other elements in an infinite group? Do you have generators in a finite group? Thanks

A set of generator of a group is a set that generates the whole group. If, for example a, b, c are a set of generators for some group, then every element of the group is of the form $a^i b^j c^j b^k c^l ...$ where you can compose any finite sequence of the letters a,b,c, in any order, inside the expressions, with any positive or negative exponents (and two consecutive letters are different).
A group may be finite or infinite, finitely generated or infinitely generated (it is, of course, always finitely generated if the group is finite, taking the whole group as a set of generators, and it may be finitely generated even if the group is infinite). Notice also that in general, two different expressions may represent the same element (this is necessary if the group is finite). If this is not the case, then the group is said to be freely generated by the given set of generators. This means that every element can be written in a unique manner as an expression of the above form.

 

[mathwonk]

generators are to a group sort of like letters of the alphabet are to the set of all words. i.e. all group elements can be made by combining generators.

except in a group usually there can be different spellings of the same word, i.e. there can be different combinations of the generators that give the same resulting group element. (maybe you haVE TO ALLOW ALSO INVERSES OF THE GENERATORS -oops).

"free groups" are ones where every different combination of some appropriate set of generators does give a different group element.one nice group is the group of all nxn matrices with entries from some familiar number system, like the reals or complexes.

a group representation is a function taking the elements of your group into elements of a matrix group, so as to preserve the multiplication, or rather so as to turn the group multiplication into matrix multiplication.

e.g. one can represent the group of permutations of n elements in the group of nxn matrices, by taking each permutation to the matrix in which either the rows or the columns are permuted by that permutation, (i forget which one makes the multiplication work out right.)

if p is a prime number, then i seem to recall that one can choose a generators for the permutation group of p elements, the set of two permutations, one an interchange of just two elements, like (12), and the other a transitive cycle, like (123...p), (and its inverse (p...321)). e.g. then the product (123...p)(12)(p...321) = (1p). i hope.

 

[Stephen Tashi]

A representation can be a set a matrices, yes?

Yes. For an infinite group, it can be an infinite set of matrices. A typical way to present an infinite set of matrices is write a matrix that has variable parameters in it - such as $\begin{pmatrix} \cos{\theta}&-\sin{\theta} \\ \sin{\theta}&\cos{\theta} \end{pmatrix}$.

Where each matrix represents an element in the group, so that the operation of the group can be reduced to simple matrix multiplication.

That statement isn't clear. You can have a representation of a group G as a group of matrices T such that G and T are different as groups. The requirement for a representation is that T will be a homomorphic image of G. So calculating multiplication in G can't necessarily be done by doing the multiplication of matrices in T. You need a representation that is an isomorphism in order to do that.

 

[resurgance2001]

Thanks for all these replies. There's a lot of very helpful instruction here. Cheers Peter