Axiom Of Choice

We have the following indirect implication of form equivalence classes:

107 \(\Rightarrow\) 308-p given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
107 \(\Rightarrow\) 62	clear
62 \(\Rightarrow\) 308-p	Maximal p-subgroups and the axiom of choice, Howard-Yorke-1987 [1987, Notre Dame J. Formal Logic

Here are the links and statements of the form equivalence classes referenced above:

Howard-Rubin Number	Statement
107:	M. Hall's Theorem: Let \(\{S(\alpha): \alpha\in A\}\) be a collection of finite subsets (of a set \(X\)) then if () for each finite \(F \subseteq A\) there is an injective choice function on \(F\) then there is an injective choice function on \(A\). (That is, a 1-1 function \(f\) such that \((\forall\alpha\in A)(f(\alpha)\in S(\alpha))\).) (According to a theorem of P. Hall (\(\)) is equivalent to \(\left \|\bigcup_{\alpha\in F} S(\alpha)\right\|\ge \|F\|\). P. Hall's theorem does not require the axiom of choice.)
62:	\(C(\infty,< \aleph_{0})\): Every set of non-empty finite sets has a choice function.
308-p:	If \(p\) is a prime and if \(\{G_y: y\in Y\}\) is a set of finite groups, then the weak direct product \(\prod_{y\in Y}G_y\) has a maximal \(p\)-subgroup.

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