diff b783308..d214c92 kaplansky_direct_finite_char2_v2.txt
diff --git a/kaplansky_direct_finite_char2_v2.txt b/kaplansky_direct_finite_char2_v2.txt index 90b18f1..e76cb65 100644 --- a/kaplansky_direct_finite_char2_v2.txt +++ b/kaplansky_direct_finite_char2_v2.txt @@ -1,52 +1,10 @@ ================================================================================ A COUNTEREXAMPLE TO KAPLANSKY'S DIRECT-FINITENESS CONJECTURE IN CHARACTERISTIC TWO - A plain-English companion version, written for curious high school readers + ================================================================================ -PURPOSE OF THIS DOCUMENT ------------------------- -The original paper (kaplansky_direct_finite_char2_original.txt, about -108 KB) is a research paper in contemporary group theory and -non-commutative algebra. It disproves a conjecture Kaplansky posed in -1970, and derives consequences that had been open in symbolic dynamics -and group theory. The audience of that paper is other group theorists -and ring theorists. - -This companion version is written for a different audience: a reader -who knows high-school algebra and a bit of linear algebra, and is -willing to think through a construction step by step, but who has not -seen modern algebra. It is meant to be read alongside the original. -Where the original proves a technical lemma, this version tries to say -WHAT the lemma says in plain words, WHY it is true, and WHY the whole -construction needs it. - -The document has these parts: - - PURPOSE OF THIS DOCUMENT (this section) - PART I -- what the conjecture says, why it is interesting, and - the basic objects (groups, fields, group rings). - PART II -- the exact statement of the main theorem and its two - consequences (nonsofic and non-surjunctive). - PART III -- the shape of the proof, as ten small steps, no details. - PART IV -- a glossary of every specialized word the paper uses. - PART V -- the original abstract and the conventions of the paper. - PART VI -- small concrete calculations a curious reader can do - with a pen, a paper, and a bit of linear algebra. - PART VII -- why this matters: connections to soficity, - surjunctivity, and cellular automata. - PART VIII -- the lemma index of the original paper. - PART IX -- exact statements of every result, verbatim from the - original, lightly re-typeset for ASCII. - -"More concise than the original" and "more complete than the original" -sound like opposites. They are not. The original is long because it -needs to justify every step to a professional reader who will check. -This companion is short because it only needs to explain the shape of -the argument to a reader who is not going to referee it. Nothing from -the original's mathematical content is dropped. Some things are -compressed; some things that the original takes for granted are -spelled out. + PART I. WHAT THIS PAPER IS ABOUT ================================================================================