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On groups and CFSG

As mentioned in the previous posts, groups can be thought of as structures describing symmetries. So, groups are just collections of transformations, that we can compose (for example first rotate an object and then reflect what we got along an axis of symmetry). And then it turns out our collection is just a set, and composition is an operation on its elements which satisfies some sensible and basic conditions – for example, we want to be able to do nothing or rewind whatever transformation we just did.

Sometimes it happens that there is a smaller group H, that lives inside another group G – taking a subset of our elements still gives us a well-behaved composition. Even better, a subgroup H might even be normal – and using it we may construct a quotient group G/H.

Simple groups are ones, that we can never make interesting quotients from - their normal subgroups are themselves or a set containing only identity (just a lonely “do nothing” element). Then quotient of a simple group by itself it’s just identity, and quotient by the identity is like leaving the group unchanged.

It often happens that the same group appears in different contexts, maybe getting differently labelled elements, or arising from a different construction. The notion of “sameness” we use in that case is being isomorphic – if two groups are isomorphic, that means we can rename their elements and the operation, and all that will result in us looking at exactly same structure. Because of that, when group theorists look at a group, they are looking at it up to isomorphism (as there is no point doing the same computation multiple times when we could just have changed labels).

As a result of thousand pages of proof, contributions of many mathematicians and half of a 20th century of work, all simple finite groups were classified (up to the “sameness” described above). The grand result that is the Classification of Finite Simple Groups (CFSG).

Sporadic groups are ones, that do not fit in into any infinite families of simple groups (groups in the same family are defined similarly, the only thing really changing is the coefficients in the construction). So, I guess some could call them a “classification bin” – but even if, I still find it incredibly interesting that there are only 26 of them (or 27 if one counts the Tits group). Doesn’t that number seem like coming out from nowhere?

(Sorry, I am still trying to find a balance of how much rigorous maths I should try to post – I am open to any comments or suggestions.)

#group theory #simple groups #research