* an alternating groups of degree ≥ 5 |
* an alternating group of degree at least 5 |
* an exceptional or twisted group of Lie type * or one of 26 left over groups known as the sporadic groups |
* an exceptional or twisted group of Lie type (including the Tits group) * or one of 26 left-over groups known as the sporadic groups |
5 of the sporadic groups were discovered by Mathieu in the 1860's and the other 21 were found between 1965 and 1975. The full list is:
The largest of the sporadic groups is the Monster group. It has 246.320.59.76.112.13317.19.23.29.31.41.47.59.71 = 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000 elements and plays a starring role in the [Monstrous Moonshine Conjectures]? of Conway and Morton which were subsequently proved by Borcherds.
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