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2 Introduction to GrpConst

This package contains three methods to construct the groups of a certain type up to isomorphism.

The Frattini Extension Method:

This method can be used to determine all soluble groups of a given order. The practicability of the method depends clearly on the chosen order. Furthermore, the efficiency of the method might be increased by restricting the construction to groups with certain properties. This is easily possible for a number of properties; for example, it is useful and straightforward to compute non-nilpotent groups only. (See Chapter The Frattini Extension Method.)

The Cyclic Split Extension Method:

The cyclic split extension method can be used to list all groups of order pn cdotq for different primes p and q which have a normal Sylow subgroup. These groups are also soluble, and hence might also be obtained by the Frattini extension method. However, the cyclic split extension method is more effective on this case. Note that this method relies on a list of groups of order pn. Moreover, the efficiency of this method depends on an effective method to compute automorphism groups of p-groups. (See Chapter The Cyclic Split Extension Method.)

The Upwards Extension Method:

The upwards extension method is the most general of the three methods. It can be used to construct all groups of a given order. However, it is the least efficient of the three methods and hence should only be used to construct non-soluble groups. Note that this method needs a list of perfect groups of order dividing the given one. (See Chapter The Upwards Extension Method.)

Furthermore, the package contains a wrap up function which combines the three methods to a general algorithm to construct the groups of a given order. (See Chapter Construction of All Groups.) Finally, there is an info class InfoGrpCon available for the functions of this share package with possible levels 1 up to 4.

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grpconst manual
Mai 2012