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Classification of finite simple groups

Computer algebra systems have been used to list all groups of order up to 2000. But classifying all finite groups is a problem considered too hard to be solved.

The classification of all finite simple groups was a major achievement in contemporary group theory. There are several infinite families of such groups, as well as 26 " sporadic groups " that do not belong to any of the families. The largest sporadic group is called the monster group. The monstrous moonshine conjectures, proved by Richard Borcherds, relate the monster group to certain modular functions.

The gap between the classification of simple groups and the classification of all groups lies in the extension problem.