quotient_group

std.algebra.quotient_group · Level L3

The quotient group G/H by a subgroup H: its elements are the cosets, labelled by their smallest element, and the product of the cosets of g and h is the coset of g·h, given as a table indexed by elements. Also whether H is normal and whether that product is well defined, the same for every choice of representatives. If H is not normal, representatives can give different cosets, and there is no quotient group. The number of cosets is the order of G/H.

(gH)(hH) = (gh)H

Signature

quotient_group(T: i64[n, n], H: f64[n]) → f64[], f64[], f64[n, n], 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 coset labels of every product, compared with those of the representatives' product in exact rational arithmetic, on all 40 test cases.
All 1,222 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

is_normal_subgroup left_cosets

Called by

quotient_ring

sha256:1253f20ce9c73a4502b56218bca84025943f97a8cdf20ddc515046ba12c6eae9

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.quotient_group · Ω:quotient_group
Pins: sha256:70776839cf09ed2bfc9a751782b0672466c3c3dea5b658747e7f764f66926df9this graph and the 6 it reaches through calls
Evidence: sha256:e842a64b9f09972b69ca91b2688df2fbb299b8082eb8e759bd2f8b67f6495f42the 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.