Maximal subgroups of direct products

A group G is simple if and only if the diagonal subgroup of G ×G is a maximal subgroup. This striking property is very easy to prove and raises the question of determining all the maximal subgroups of G , where G denotes the direct product of n copies of G . The first purpose of this note is to answer completely this question. We show in particular that if G is perfect, then any maximal subgroup of G is the inverse image of a maximal subgroup of G for some projection G → G on two factors. If G is finite, we let m(G) be the number of maximal subgroups of G . If G = Cp is cyclic of prime order p , then m(C p ) = p − 1 p− 1 , so that m(C p ) is an exponential function of n . It follows easily that m(G) grows exponentially if G is not perfect. In contrast, when G is perfect, the fact that any maximal subgroup of G comes from G implies that m(G) is a quadratic polynomial in n . We give in fact an explicit formula for m(G) (in terms of numbers depending only on G ). The minimal number d(H) of generators of a finite group H highly depends on the number of maximal subgroups of H . For instance, if there is only one maximal subgroup, then H is cyclic (of prime power order) and d(H) = 1 . So it is not surprising that the above results imply that d(G) behaves differently depending on whether or not G is perfect. It turns out that d(G) grows logarithmically if G is perfect and linearly otherwise. This result is due to Wiegold [W1, W2] and we give here a new proof based on our study of maximal subgroups. There is a general procedure for finding the maximal subgroups M of a finite group, due to Aschbacher and Scott [A-S] (see also the work of Gross and Kovacs). Although their assumption that M should be core–free could be realized in our case, our elementary methods do not make it necessary and this little note does not depend on their important work. We should perhaps apologize for the fact that this note is so elementary that it could be taught to undergraduates, but, as Schönberg said, there are still many musics to be written in C major.