What is computational group theory?

What is the difference between computational group theory and group theory?

Is it an active area of the mathematical research currently?

What are some of the most interesting results?

What is the needed background to study it?


There is a nice survey of the subject area available in pdf: Survey: Computational Group Theory, which while somewhat dated, gives a nice introduction to the field and provides some historical insights.

Here's a very nice Introduction to Computational Group Theory. It's a brief but fascinating survey by Ákos Seress, published in the AMS Notices (1997: 06).

See also, of course, Wikipedia: Computational Group Theory. It's not a very lengthy entry, but there are nice references provided, and links that expand a bit more on what is discussed. Wikipedia mentions two computer algebra systems: GAP and Magma, which each have links to learn more. They are incredibly useful, powerful, and time-saving systems that enriches the study of group theory. GAP is freely available from its website, and also as part of the SAGE system, which is free to download as well, but can also be used online.

References include:

  • Derek F. Holt $\dagger$, Bettina Eick, Bettina, Eamonn A. O'Brien, "Handbook of computational group theory", Discrete Mathematics and its Applications (Boca Raton). Chapman & Hall/CRC, Boca Raton, FL, 2005.

  • Charles C. Sims, "Computation with Finitely-presented Groups," Encyclopedia of Mathematics and its Applications, vol 48, Cambridge University Press, Cambridge, 1994.

  • Ákos Seress, "Permutation group algorithms", Cambridge Tracts in Mathematics, vol. 152, Cambridge University Press, Cambridge, 2003.

$\dagger$ Note that author Derek F. Holt is a regular contributer on Math.SE, and has provided a nice link in a comment below!

  • 1
    $\begingroup$ Nice Observation $\endgroup$ – mrs Jun 10 '13 at 1:16
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    $\begingroup$ For interesting results, you could see also my brief answer to math.stackexchange.com/questions/408403. $\endgroup$ – Derek Holt Jun 10 '13 at 8:55
  • $\begingroup$ @amWhy: Wouldn't it be more precise to say e.g. "GAP is freely available from its website, and also as part of the SAGE system, which is free to download as well, but can also be used online"? IMHO, saying "GAP is freely available, usually as part of the SAGE system" is a bit misleading since one need not to download SAGE to use GAP... $\endgroup$ – Alexander Konovalov Jun 10 '13 at 21:20
  • $\begingroup$ @AlexanderKonovalov Corrected. Thank you. $\endgroup$ – Namaste Jun 10 '13 at 21:40
  • $\begingroup$ @amWhy: Thank you! My +1 was already there since this morning :) $\endgroup$ – Alexander Konovalov Jun 10 '13 at 21:43

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