- For each element a in G there is an element ā in G with ā ° a = a ° ā = 1
(the element ā is called the inverse element of a). - For all elements a, b and c in G it is true that a ° (b ° c) = (a ° b) ° c (this
is called the associative law).
Nearly all basic groups can be classified into known classes. The complete
classification, called ‘the enormous theorem’, was announced by Daniel
Gorenstein in 1983 and was arrived at through the accumulated work of 30
years’ worth of research and publications by mathematicians. It is an atlas of all
known groups. The basic groups fall into one of four main types, yet 26 groups
have been found that do not fall into any category. These are known as the
sporadic groups.
The sporadic groups are mavericks and are typically of large order. Five of the
smallest were known to Emile Mathieu in the 1860s but much of the modern
activity took place between 1965 and 1975. The smallest sporadic group is of
order 7920 = 2^4 × 3^2 × 5 × 11 but at the upper end are the ‘baby monster’ and
the plain ‘monster’ which has order 2^46 × 320 × 5^9 × 7^6 × 11^2 × 13^3 × 17 × 19
× 23 × 29 × 31 × 41 × 47 × 59 × 71 which in decimal speak is around 8 ×
1053 or, if you like, 8 with 53 trailing zeros – a very large number indeed. It can
be shown that 20 of the 26 sporadic groups are represented as subgroups inside
the ‘monster’ – the six groups that defy all classificatory systems are known as
the ‘six pariahs’.
Although snappy proofs and shortness are much sought after in mathematics,
the proof of the classification of finite groups is something like 10,000 pages of
closely argued symbolics. Mathematical progress is not always due to the work of
a single outstanding genius.