GroupTheory`
GroupTheory`

GTProductGroupQ

GTProduct[group, group1, group2]

gives True if group is a product group of the groups group1 and group2, and False otherwise.

Details and Options

  • First it is checked, if group1 and group2 are subgroups of group . Furthermore it is tested that group1 and group2 have only the element in common. In an next step it is tested, if the product of group1 and group2 is really a group. After the check of all suppositions it is finally checked if the product is a product group or a semi-direct product group.
  • The following options can be given:
  • GOVerbose FalseControls the input validation
    GOFast GOFastValueControls the output of additional information
  • See: W. Hergert, M. Geilhufe, Group Theory in Solid State Physics and Photonics. Problem Solving with Mathematica, chapter 3.4.

Examples

open allclose all

Basic Examples  (1)

The group will be installed and checked which subgroups and invariant subgroups exist.

Now the construction of as a product can be checked.

Find the positions of the invariant subgroups in sgr.

Which product of subgroups gives the group ?

Options  (2)

GOVerbose  (1)

Sometimes it will be interesting which kind of product is formed by the two groups, or why the two groups do not give the expected product group.

The group is constructed from two of its invariant subgroups.

Here grp2 is an invariant subgroup, but grp1 is not.

The product of the two subgroups does not lead to .

GOFast  (1)

Controls the input validation.