A descent principle in modular subgroup arithmetic

We establish and comment on a surprising relationship between the behaviour modulo a prime p of the number image of index n subgroups in a group image, and that of the corresponding subgroup numbers for a subnormal subgroup of p-power index in image. One of the applications of this result presented here concerns the explicit determination modulo p of image in the case when image is the fundamental group of a finite graph of finite p-groups. As another application, we extend one of the main results of the second authorʹs paper (Forum Math, in press) concerning the p-patterns of free powers G*q of a finite group G with q a p-power to groups of the more general form H*G*q, where H is any finite p-group.