7.87 Set Functions for Right Cosets

Right cosets are domains, thus all set theoretic functions are applicable to cosets (see chapter Domains). The following describes the functions that are implemented especially for right cosets. Functions not mentioned here inherit the default function mentioned in the respective sections.

More technically speaking, all right cosets of generic groups have the operations record RightCosetGroupOps, which inherits its functions from DomainOps and overlays the components mentioned below with more efficient functions.

In the following let C be the coset U * u.

Elements( C )

To compute the proper set of elements of a right coset C the proper set of elements of the subgroup U is computed, each element is multiplied by u, and the result is sorted.

IsFinite( C )

This returns the result of applying IsFinite to the subgroup U.

Size( C )

This returns the result of applying Size to the subgroup U.

C = D

If C and D are both right cosets of the same subgroup, = returns true if the quotient of the representatives lies in the subgroup U, otherwise the test is delegated to DomainOps.=.

h in U

If h is an element of the parent group of U, this returns true if the quotient h / u lies in the subgroup U, otherwise the test is delegated to DomainOps.in.

Intersection( C, D )

If C and D are both right cosets of subgroups U and V with the same parent group the result is a right coset of the intersection of U and V. The representative is found by a random search for a common element. In other cases the computation of the intersection is delegated to DomainOps.Intersection.

Random( C )

This takes a random element of the subgroup U and returns the product of this element by the representative u.

Print( C )

A right coset C is printed as (U * u) (the parenthesis are used to avoid confusion about the precedence, which could occur if the coset is part of a larger object).

C * v

If v is an element of the parent group of the subgroup U, the result is a new right coset of U with representative u * v. Otherwise the result is obtained by multiplying the proper set of elements of C with the element v, which may signal an error.

v * C

The result is obtained by multiplying the proper set of elements of the coset C with the element v, which may signal an error.

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