Characteristics quotients of the Universal p-Frattini cover of a finite group

One case can guide the group theory behind the construction of M(odular) T(ower)s: The composition of Chebychev polynomials. Anyone who has taught 1st year calculus has had this arise for integrating powers of trigonometric polynomials.

The analog depends on knowing that there exists a natural sequence of covers with p group kernels of any finite group G with order divisible by p. The key being the first construction step in that sequence. The analogy works well if G is p-perfect, where it produces spaces for which we can formulate conjectures on various ways they are similar to modular curves.

[§I, rims-rev.html] (especially §I.3) describes how to compute – as precisely as we know in general – the rank of the module