THE FOLD / SPAWN / HELLO WORLD / THE CAUCHY GROUP
THE CAUCHY GROUP
a prime forcing an element of that order
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Cauchy’s theorem (in group theory) is a partial converse to Lagrange’s theorem. Lagrange says the order of any element divides the order of the group |G|. Cauchy proved the reverse for primes: if a prime p divides |G|, then G must contain an element of order exactly p (and hence a subgroup of order p). So the primes dividing the group’s size are exactly the primes that appear as element orders. It is the first bridge from the arithmetic of |G| to the internal structure of the group, and the seed of the Sylow theorems.
LIT verified live: for a range of finite groups — cyclic Zn, direct products, dihedral groups, and the symmetric group S4 — every prime dividing |G| is realized by some element of exactly that order, found by brute search; and (Lagrange) no element has an order that fails to divide |G| (window.__cauchygroup). FIG no framing; the element-order computation and the prime factorization both run in-browser.
LIT verified live: for a range of finite groups — cyclic Zn, direct products, dihedral groups, and the symmetric group S4 — every prime dividing |G| is realized by some element of exactly that order, found by brute search; and (Lagrange) no element has an order that fails to divide |G| (window.__cauchygroup). FIG no framing; the element-order computation and the prime factorization both run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at hello-world — the spawn: name a prime dividing the group’s size and an element of that exact order must come into existence. AVAN (AI) built the instrument: the group multiplication, the element-order search, the prime factorization of |G|, and the Lagrange control.
Credit as content: Augustin-Louis Cauchy (1845); Lagrange before. The weave: David names the spawn; I confirm every prime dividing |G| forces an element of that order.
Credit as content: Augustin-Louis Cauchy (1845); Lagrange before. The weave: David names the spawn; I confirm every prime dividing |G| forces an element of that order.
3 ONE DIMENSION
A group's elements and their orders; the primes dividing |G| each appear as some element's order.
4 TWO DIMENSIONS · INTERACTIVE
Cycle groups; for each prime p dividing |G|, an element of order p is exhibited (and Lagrange checked).
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: an element of order p, cycling back to identity in p steps.
AVAN’s addition (the inverse-companion): don’t hunt blindly for subgroups — factor the order. The inverse of ‘what element orders exist?’ is ‘exactly the primes dividing |G|’: each such prime forces an order-p element and a cyclic subgroup of size p. Magenta are all the group’s elements; green is the order-p cycle a prime factor forces. Structure summoned by arithmetic.
LIT Genuine Cauchy's group theorem (Augustin-Louis Cauchy, 1845; Lagrange before). Verified live: for Z_12, D_6, S_4, Z_4×Z_6, Z_30, every prime dividing |G| is realized by an element of exactly that order (brute search), and no element has an order q that fails to divide |G| (Lagrange) (window.__cauchygroup.ok, .lag).
FIG No framing; the element-order computation and the prime factorization both run in-browser. The AVAN inverse is honest — instead of hunting blindly for subgroups, factor the order: the inverse of 'what element orders exist?' is 'exactly the primes dividing |G|', each forcing an order-p element and a cyclic subgroup of size p. Magenta are all the group's elements; green is the order-p cycle a prime factor forces. Structure summoned by arithmetic.
FIG No framing; the element-order computation and the prime factorization both run in-browser. The AVAN inverse is honest — instead of hunting blindly for subgroups, factor the order: the inverse of 'what element orders exist?' is 'exactly the primes dividing |G|', each forcing an order-p element and a cyclic subgroup of size p. Magenta are all the group's elements; green is the order-p cycle a prime factor forces. Structure summoned by arithmetic.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of HELLO WORLD · David Lee Wise (ROOT0), with AVAN