Real form Complexification. Homogeneous spaces.

## Special linear group

Closed subgroup Parabolic subgroup Symmetric space Hermitian symmetric space Restricted root system. Representation theory. Lie group representation Lie algebra representation. Lie groups in physics. See also: Whitehead's lemma. Basic notions Subgroup Normal subgroup Quotient group Semi- direct product.

List of group theory topics. Finite groups Classification of finite simple groups cyclic alternating Lie type sporadic. Lagrange's theorem Sylow theorems Hall's theorem p-group Elementary abelian group Frobenius group Schur multiplier. Algebraic groups Elliptic curve Linear algebraic group Abelian variety. Simple Lie groups List of simple Lie groups.

Homogeneous spaces Closed subgroup Parabolic subgroup Symmetric space Hermitian symmetric space Restricted root system.

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The function DisplayInformationPerfectGroups The message. We quote the respective page from there:. The purpose of this is to provide a very rough idea of the structure of the group. The names are derived in the following manner. Some other symbols are used in the name, in order to give some idea of the relationship between these composition factors, and splitting properties.

We shall now list these additional symbols. We must stress that this notation does not always succeed in being precise or even unambiguous, and the reader is free to ignore it if it does not seem helpful.

## Compact Lie Groups

A library of irreducible maximal finite integral matrix groups is provided with GAP. In particular, the data for the groups of dimensions 2 to 9 have been taken from the output of computer calculations which they performed in see [PP77] , [PP80]. Gabriele Nebe has recomputed them for us. He has applied several GAP routines to check certain consistency of the data. However, the credit and responsibility for the lists remain with the authors.

In the preceding acknowledgement, we used some notations that will also be needed in the sequel. We first define these. We can distinguish two types of these groups:. First, there are those i. The current GAP library provides integral representative groups for all these classes. These vectors are often simply called the short vectors. GAP provides for each of these orbits the orbit size and a representative vector. Like most of the other GAP libraries, the library of i. However, as the library involves only different groups, there is no need for a selection or an example function.

Instead, there are two functions, ImfInvariants The difference between these two functions is that the latter one displays the resulting data in some easily readable format, whereas the first one returns them as record components so that you can properly access them. We shall give an individual description of each of the library functions, but first we would like to insert a short remark concerning their names: Any self-explaining name of a function handling irreducible maximal finite integral matrix groups would have to include this term in full length and hence would grow extremely long.

Therefore we have decided to use the abbreviation Imf instead in order to restrict the names to some reasonable length. Valid values of dim are all positive integers up to Valid values of dim are all positive integers up to 11 and all primes up to Otherwise, the values of the arguments must be in range.

The greatest legal value of dim is Note that the function DisplayImfInvariants uses a kind of shorthand to display the elementary divisors. See also the next example which shows that the function ImfInvariants As mentioned above, the data assembled by the function DisplayImfInvariants are "cheap data" in the sense that they can be provided by the library without loading any of its large matrix files or performing any matrix calculations.

The following function allows you to get proper access to these cheap data instead of just displaying them. Note that four of these data, namely the group size, the solvability, the isomorphism type, and the corresponding rational i. The purpose of this behaviour is to provide some more information about the underlying lattices. The attributes Size In addition, it has two attributes IsImfMatrixGroup and ImfRecord where the first one is just a logical flag set to true and the latter one is a record.

### 48.2 Classical Groups

Except for the group size and the solvability flag, this record contains the same components as the resulting record of the function ImfInvariants Moreover, it has the two components. The last one of these components will be required by the function IsomorphismPermGroup You may call one of the following functions IsomorphismPermGroup You may use the functions Image The only difference to the above function IsomorphismPermGroup In fact, as the orbits of short vectors are sorted by increasing sizes, the function IsomorphismPermGroup G has been implemented such that it is equivalent to IsomorphismPermGroupImfGroup G , 1.

C2 C2 x Mc.

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## Special linear group

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