Sylow's Theorem and the arithmetic of binomial coefficients

Abstract

We present a result on the existence and the number of subgroups of any given prime-power order containing an arbitrarily fixed subgroup in a finite group (see also [2]). Our proof is an extension of Krull's generalization ([1],1961)of Sylow's theorem, which leads us to consider a new concept (the conditioned binomial coefficient) of independent combinatorial interest.