Frank Lübeck |

Note: the structures of all given groups are already found by several people. Nevertheless, we leave these examples here, since they may still be useful.

# What can you say about these matrix groups?

Here are some examples of matrix groups over finite fields, generated by Frank Lübeck and Jürgen Müller. Their construction was motivated by a request of several people from the*Matrix Group Recognition Project*. They asked for challenge input for their programming systems. Clearly, anybody is invited to investigate these groups.

For each of the example matrix groups, can you find out one or several of the following topics?

- The order of the group.
- A composition series or the set of composition factors with or without their multiplicities.
- Further details of the group and subgroup structure.
- A construction of an isomorphic group.

Tell us if you would like to see some "smaller" or "larger" examples.

**Acknowledgement.** We would like to thank Peter Neumann for a
useful discussion about some of our examples.

Frank Frank.Luebeck@Math.RWTH-Aachen.De

Jürgen Juergen.Mueller@Math.RWTH-Aachen.De

PS: Peter Neumann offers another incentive to look at these examples! He
wrote to us:

I'll be happy to offer a king-size Mars-bar for the first
correct and reasonably full solution to each of your challenges.

## The Examples

Example 1Example 2

Example 3

Example 4

Example 5

Example 6

Example 7

Example 8

Example 9

Example 10

Example 11

Example 12

All examples as GAP file

Further examples constructed by Rob Wilson and Richard Parker can be found here.

Last updated: Tue Apr 5 14:44:56 2016 (CET)