Implications

We have the following indirect implication of form equivalence classes:

335-n \(\Rightarrow\) 373-n
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
335-n \(\Rightarrow\) 288-n Some equivalents of \(AC\) in algebra, Keremedis, K. 1996b, Algebra Universalis
288-n \(\Rightarrow\) 373-n clear

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

Howard-Rubin Number Statement
335-n:

Every quotient group of an Abelian group each of whose elements has order  \(\le n\) has a set of representatives.

288-n:

If \(n\in\omega-\{0,1\}\), \(C(\aleph_0,n)\): Every denumerable set of \(n\)-element sets has a choice function.

373-n:

(For \(n\in\omega\), \(n\ge 2\).) \(PC(\aleph_0,n,\infty)\): Every denumerable set of \(n\)-element sets has an infinite subset with a choice function.

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