Yesterday, my friend and ex-colleague Dan Hughes emailed a group of his friends to explain that he had established that earth had been visited in antiquity by alien group theorists. How else to explain that the number of minutes in an hour, the number of hours in a week, and the number of degrees in a circle, are the orders of the first three non-abelian finite simple groups?
Looking for further evidence, I noticed that the number of minutes in a fortnight is the smallest number for which there exist two non-isomorphic finite simple groups.
But the exchange reminded me of an observation I made as a student about the small finite simple groups.
There is, not surprisingly, a lot of numerology associated with the first three primes, 2, 3 and 5. For example, they are the orders of the rotational symmetries of a regular dodecahedron. But if we include the next two primes as well, further interesting things emerge.
2, 3 and 5 are Sophie Germain primes: this means that 2.2+1=5, 2.3+1=7, and 2.5+1=11 are also primes (indeed they complete the tally of the first five primes).
But let us return to group theory, I believe (though I cannot recall the citation) that it was Galois who first realised that the linear fractional group PSL(2,p) over the integers mod p, for prime p, has a subgroup of index p+1. This is usually the smallest index of a subgroup of this group; but, precisely for the first five primes, there is an unexpected subgroup of index p. In fact, the subgroups of index p in the last three of these groups (for p=5, 7 and 11) are the polyhedral groups A4, S4, and A5. (And these lift to the binary polyhedral groups in the special linear group SL(2,p), as I discussed in an earlier post.)
(As an aside, Galois invented finite fields, and studied linear fractional groups over fields of prime order; but I do not believe he made the obvious synthesis. He may have been amused to realise that PSL(2,9) has a subgroup of index 6.)
But on to what I realised in my student days. This relates to the important results of Philip Hall on finite soluble groups (or solvable groups, if you are divided from me by our common language). Hall studied what are now called Hall π-subgroups of a group G, where π is a set of primes; such a subgroup is one whose order is divisible only by primes in π, and whose index in G is divisible only by primes outside π. Thus, if π consists of a single prime p, then we are talking about a Sylow p-subgroup of G; by Sylow’s theorem, such subgroups exist, and are all conjugate.
Among much else, Hall proved that Hall π-subgroups of finite soluble groups exist, and are conjugate (and hence isomorphic), for all possible sets of primes; and that this property characterizes soluble groups among all finite groups.
So examples where Hall’s properties fail will be insoluble, and presumably finite simple groups are good places to look. We find the following:
- The smallest group which fails to have a Hall π-subgroup for some set π of primes is PSL(2,5).
- The smallest group which has non-conjugate Hall π-subgroups, for some set π of primes, is PSL(2,7).
- The smallest group which has non-isomorphic Hall π-subgroups, for some set π of primes, is PSL(2,11).