On 12 November 2016 at 13:06, Martin Baker <ax87...@martinb.com> wrote:
> On 12/11/16 17:37, Bill Page wrote:
>>
>> I think I know what you mean however in FriCAS % always represents a
>> domain - not an element of a domain. In the category 'Group' %
>> represents a domain whose operations satisfy group axioms. Perhaps it
>> is unexpected that so few domains in FriCAS satisfy Group but the
>> requirement is that every value of this type must have a
>> multiplicative inverse, literally an inverse for the operation *.
>
>
> Bill,
>
> Should I have said Rep rather than % ?
>
> In one case Rep holds something which represents the whole group. In the
> other case Rep holds an element of that group.
>
No, that's definitely worse. Rep is something internal - just an
implementation detail. What you are talking about are just the values
of a given type.
Values of types (domains) that satisfy 'Group' such as
(1) -> FreeGroup(Symbol) has Group
(1) true
Type: Boolean
(2) -> Permutation(Symbol) has Group
(2) true
Type: Boolean
have values that are elements of a group. For example:
(3) -> p1:=[a::Symbol,b::Symbol,c::Symbol]::Permutation(Symbol)
(3) (a b c)
Type: Permutation(Symbol)
(4) -> p2:=[a::Symbol,c::Symbol,b::Symbol]::Permutation(Symbol)
(4) (a c b)
Type: Permutation(Symbol)
But
(5) -> PermutationGroup(Symbol) has Group
(3) false
Type: Boolean
so the elements of PermutationGroup(S) are not elements of a group,
rather they are (representations) of a group generated by a list of
generators. For example:
(6) -> pg1:=[p1,p2]::PermutationGroup(Symbol)
(6) <(a b c),(a c b)>
Type: PermutationGroup(Symbol)
In principle I suppose that you could define a domain such as
PermutationGroupByGenerators(S, List Permutation(S))
the values of which would actually be elements of a group.
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