On Wed, Mar 24, 2021 at 3:59 PM slelievre wrote:
>
> 2021-03-24 14:53:06 UTC, Matt:
>>
>> How can I calculate the direct product of two groups and/or the direct sum?
>> For example given:
>>
>> G.=FreeGroup()
>> H=G.quotient([x*y])
>>
>> how do I calculate H+H and HxH?
>
>
> Having defined `H`, you can type
>
> H.
>
> then hit the TAB key to see what methods are available.
>
> This reveals the methods `cartesian_product`
> and `direct_product`, are they what you need?
>
> ```
> sage: G. = FreeGroup()
> sage: H = G.quotient([x*y])
>
> sage: H.cartesian_product(H)
> The Cartesian product of
> (Finitely presented group < x, y | x*y >,
> Finitely presented group < x, y | x*y >)
>
> sage: H.direct_product(H)
> Finitely presented group < a, b, c, d |
> a*b, c*d, a^-1*c^-1*a*c, a^-1*d^-1*a*d,
> b^-1*c^-1*b*c, b^-1*d^-1*b*d >
> ```
careful here, FreeGroup is not Abelian, you'll want to add commuting relations
for all its generators.
You might prefer AbelianGroup() to begin with.
>
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