Maximal p-subgroups and the axiom of choice (Q1097269)

Consider the following statements: S(p): Every group has a maximal p- subgroup. Sp(p): If G is the weak direct product of symmetric groups then G has a maximal p-subgroup. Below are some representative theorems from this paper. Theorem. For any prime p, S(p) is equivalent to the axiom of choice. Theorem. If \(p_ 1\neq p_ 2\) are primes, \((SP(p_ 1)\) and \(SP(p_ 2))\) is equivalent to the axiom of choice for sets of finite sets. Theorem. (In Zermelo- Fraenkel set theory) If p is a prime, SP(p) does not imply the axiom of choice for sets of p-element sets.