Today in class on finite group theory our professor taught us Mathieu groups and so we dealt with Steiner system and similar.

He said from here on you can pursue computational group theory and start using computers to do stuff with bigger groups, like as to find orders of Mathieu groups or to check $S(5,6,12)$ is a Steiner system etc etc. He said that from now many proofs will require mathematical softwares like GAP etc).

As he is old and he is not so much into softwares or computational group theory, can somebody recommend to me where to begin, and what softwares I will need to study computational group theory? Where can I get them and a good book.

I am also not much acquainted with any mathematical softwares, but would like to use the ones which are useful for finite groups self-study.

  • $\begingroup$ I think GAP is quite good place to start. gap-system.org In this site, you can find program itself and manuels tutorials etc. $\endgroup$ – mesel Sep 24 '14 at 19:16
  • $\begingroup$ are there books on this subject? good ones. $\endgroup$ – Bhaskar Vashishth Sep 24 '14 at 19:18
  • $\begingroup$ I do not know whether there is a published book, you can check gap-system.org/Doc/doc.html $\endgroup$ – mesel Sep 24 '14 at 19:22
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    $\begingroup$ I do not think that "As he is old ..." is an appropriate motivation of this question. $\endgroup$ – Alexander Konovalov Sep 24 '14 at 20:22

Computer algebra systems, such as GAP, have already been mentioned in the comments for software. For books, there is the Handbook of Computational Group Theory (Amazon). It's very good; good enough that I have two copies, for home and the office, despite the price. The pace might be a bit brisk in places for an introduction, though. But it covers a lot of ground.

There are more specialised works, as well. Sims' book covers computing with finitely presented groups, and is at about the same level as the Handbook. A book by Butler is quite good at a very introductory level, but is specific to permutation groups. Nevertheless, these are two very important areas in computational group theory, each worth studying in its own right.

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    $\begingroup$ Yes, the Handbook of Computational group Theory is definitely to be recommended (but I am not competely unbiased). The book by Sims is more formal with more rigorous proofs of correctness of algorithms, etc, but it is also very good on the technical details of fundamental algorithms such as coset enumeration. There is also a more specialized book on permutation group algorithms, with accurate complexity analysis by Seress. $\endgroup$ – Derek Holt Sep 24 '14 at 20:04
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    $\begingroup$ Also, Alexander Hulpke prepared Notes for a course in Computational Group Theory and Instructor's Manual for GAP. You may find them here. $\endgroup$ – Alexander Konovalov Sep 24 '14 at 21:50

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