element_of_order

std.algebra.element_of_order · Level L3

Whether a finite group has an element of order d, as 1 or 0, and the first one (0 when there is none). Cauchy's theorem says one exists for every prime p dividing |G|; for a composite divisor d there may be none (Z₂ × Z₂ has no element of order 4).

∃g: ord(g) = d

Signature

element_of_order(T: i64[n, n], d: f64[]) → f64[], f64[]

Structure

The function as NOVA stores it: one box per input, operation and output, and arrows that carry values. A double border marks another library function this one runs — called once, or by Scan once per element; select it to open that function.

input
operation
constant
call
output

Verification

Signature proven by NOVA’s shape solver, for every size.
Equal to the reference the first element whose order is d in exact rational arithmetic, on all 40 test cases.
All 80 float64 results are exact: the error is zero.
Interpreter and NumPy backend return bit-identical results.
At float32 too: the 40 test cases rounded to float32, and 40 of them exact, with zero error.

Accuracy in detail

correctly rounded (the float64 nearest the exact value): 100%
bit-equal to the NumPy formula in float64: 100%
largest error, in units in the last place: 0

Identity

Calls

element_orders

Called by

—

sha256:690340f495b4239496494ae7de1619e3430aa9e6a33256f09a2591bbdb7f1090

The semantic hash of the graph. It changes when the program changes, and never when only its documentation does.

Control handle

Symbol: Ω:std.algebra.element_of_order · Ω:element_of_order
Pins: sha256:19a666a2956a2155c3749c49d3090add649903243d42647f094d04f6c17a9facthis graph and the 3 it reaches through calls
Evidence: sha256:ab5e95e4e6a1ae31e48f2295ca7221f57ae3b7eebbb86a08ceaff40c7fea5744the hash of its verification record
Needs: no capability: a pure function

Through NOVA’s control layer, the symbol launches this function only while the program still matches what it pins: a change to this graph, or to any graph it reaches, needs a migration first.