In this file we define the complement of a subgroup.
Main definitions #
is_complement S Twhere
Tare subsets of
Gstates that every
g : Gcan be written uniquely as a product
s * tfor
s ∈ S,
t ∈ T.
Tis a subset of
Gis the set of all left-complements of
T, i.e. the set of all
S : set Gthat contain exactly one element of each left coset of
Sis a subset of
Gis the set of all right-complements of
S, i.e. the set of all
T : set Gthat contain exactly one element of each right coset of
Main results #
is_complement_of_coprime: Subgroups of coprime order are complements.
T are complements if
(*) : S × T → G is a bijection.
This notion generalizes left transversals, right transversals, and complementary subgroups.