Maths - Group Theory - Linear Group Types

Represention Theory

Represention theory is a specialisation of group theory that is restricted to vector spaces.

permutation

Classification of Groups

categorise groups

Conjugacy Classes

There is a 1:1 correspondence between transitive actions and conjugate classes.

Conjugacy classes of subgroups Transitive actions of G
collection of subgroups conjugate to some given subgroup.

Transitive = one orbit

Any action = sequence of transitive actions.

Actions

We can look at the concepts that we used for permutations, now that we are restricting ourselves to linear maps:

Stabiliser

Subgroup of G consisting of all elements h such that:

h x = x

h is the stabiliser of x

Orbit

 

Centre

 

Mat(n,F)

May be any n×n matrix containing R, C, H or O elements.

The definitions don't make it totally clear whether Mat(2,C) is any 2×2 matrix whose elements are C or is it the matrix algebra over the centre of C
From Wikipedia: "The term center or centre is used in various contexts in
abstract algebra to denote the set of all those elements that commute with
all other elements".

SL(n, F) The Special Linear Group

Consisting of n×n matrices where each element is of type 'F' with determinant =1.

 


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see also:

 

Correspondence about this page

Book Shop - Further reading.

Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.

flag flag flag flag flag flag Symmetry and the Monster - This is a popular science type book which traces the history leading up to the discovery of the largest symmetry groups.

Terminology and Notation

Specific to this page here:

 

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