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512000 = Universal Frattini Extension

Item posted by Daniel Wyckoff <deluger> on Mon 15 Feb 2021 09:56:27 PM UTC.

Anyone curious about the alchemical nature of formalization, our real reward, knowing we're putting digital spaces at ease, might be amused by my tarball-size arithmetic that led to 1280000 = 768000 + 512000.  768000 = 512000 + 256000.  512000 = 256000 + 256000.  Those 4 numbers appeared as tar-ball sizes, in some Hermetic arithmetic of improvised music text files.  the 1280000 is obviously Boolean, maybe free Boolean automorphism group -- covered by Sikorski's Extension Criterion.  The 768000 is obviously partial algebras.  But, since the automorphism group of partial algebras would be affected digitally by the formalization, we would need to connect 512000 to it, hopefully, and this can be done when you realize that the interesting digital alchemy stems from quotients of automorphism groups of partial algebras by automorphism groups of models of their subsignatures.  The interesting quotients would be elementary abelian, since those groups' automorphism groups are closely related to themselves, either isomorphic, or something nice, and they arise from quotients (like from symmetric groups) that also have nice isoautomorphic properites!  Now a Frattini extension is an extension of an elementary abelian group into the Frattini subgroup of a group G by an arbitrary factor Q, and we need that to preserve digital magic!  So, to summarize -- formalizing partial algebras induces automorphisms that can be modelled by Frattini extensions.  We have the math, and by formalizing, we give interfaces to the digital spaces' other side.  And now how about 256000?  That would have to be related to some original research of mine regarding quotients of automorphism groups of partial algebras by automorphism groups of superpositionally (via function algebras) related "extensions" -- they're also Frattini elements in some key group, perhaps the quotient of the symmetric group on a given carrier of an algebra by its automorphism group of that carrier.  More fun, less time -- and also embedded devices are still on as I try to interface all of this math with the simplest electronics that exhibit anomalies.  So, the next big theorem to formalize will be theorem B.2. in Appendix B in Doerk/Hawkes "Finite Soluble Groups," that constructs a universal Frattini extension.  Only then will I think about formalizing my original stuff.  I'm almost done with the Recursion Theorem from Reichel's text, but it's very subtle and no more important than getting my finite types and finite group theory library up.  Balance!

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