Add definitions of vector spaces and algebras to the GAP Tutorial?

[lhsoicher]

UPDATE: almost all of the points below have been taken care of, all that remains is to add mathematically complete definitions for vector spaces and algebras to section 6.1 and 6.2 in the tutorial.

UPDATE2: a concrete suggestion is in the comments, someone just has to add it in

Suggested fixes for "GAP -- A Tutorial Release 4.11.0, 29-Feb-2020"

On page 41, in the first sentence after the first Example,
"its first three generators are trivial, meaning that the first three
generators" should be "its last three generators are trivial, meaning that
the last three generators".

Also on page 41, in the paragraph before the last Example, the definition
of simple group could be more precise, say "The factor group f is a simple group,
i.e., it is a non-trivial group whose only normal subgroups are its trivial subgroup
and itself."

On page 43, 5 lines before the last Example, delete "always assumes".

Also on page 43, on line -4, "fixpoint" should be "fixed point".

On page 44, on the second line after the first Example, to be more
precise, "given as sorted lists" should be "given as strictly sorted
lists".

On page 46, line -4, You write "without constructing external sets",
but I don't think you ever say what you mean by an "external set".

On page 50, second line after the second Example, change "the conjugation"
to either "the conjugation action" or just "conjugation".

In the last Example on page 50, somehow the output of a block is wrong.
It should be:

gap> Set(blocks[1]);
[ (1,2)(3,4)(5,6)(7,8), (1,3)(2,4)(5,8)(6,7), (1,4)(2,3)(5,7)(6,8),
(1,5)(2,6)(3,8)(4,7), (1,6)(2,5)(3,7)(4,8), (1,7)(2,8)(3,6)(4,5),
(1,8)(2,7)(3,5)(4,6) ]

Please double-check this. (It is consistent with the remainder of Section 5.3,
unlike the current given output.)

On page 54, in the first line of Section 5.5, "to calculate in" should be
"to calculate is in".

On page 57, at the beginning of Section 6.1, you should give a
mathematically complete definition of a vector space over a field F.

On page 60, at the beginning of Section 6.2, you should give a
mathematically complete definition of an algebra.

On page 70, in the last sentence before Section 7.7, change
"return a list" to "return an immutable list".

On page 71, lines 1-2, "SylowSubgroup( G )" should have another argument,
say, "SylowSubgroup( G, 2 )".

[fingolfin]

Most of this was taken care of by @hulpke in PR #4111 resp. 9b20de7, "apart from adding full definitions in sections 6.1 and 6.2".

Volunteers to take care of that (i.e., adding in the missing definitions) are welcome :-)

[lhsoicher]

I suggest the following for the definition of a vector space at the beginning of Section 6.1:

A vector space V over a field F is an (abelian) additive group
that is closed under scalar multiplication by elements in F,
such that 1v=v, a(bv)=(ab)v, a(v+w)=av+aw,
and (a+b)v=av+bv, for all a,b \in F and all v,w \in V.

For the definition of an algebra at the beginning of Section 6.2, I believe the only change required
is to replace "If a multiplication is defined" by "If a bilinear multiplication is defined".

[lhsoicher]

Hi @hulpke @fingolfin

I'll try again with my suggestions for the definitions of a vector space at the beginning of Section 6.1 and
for an algebra at the beginning of Section 6.2. Hopefully this time the xml tags for GAPDoc won't be automatically removed.

A <E>vector space</E> <M>V</M> over a field <M>F</M> is an (abelian) additive group
that is closed under scalar multiplication by elements in <M>F</M>, such that 
<M>1v=v</M>, <M>a(bv)=(ab)v</M>, <M>a(v+w)=av+aw</M>, and <M>(a+b)v=av+bv</M>, 
for all <M>a,b \in F</M> and all <M>v,w \in V</M>.

For the definition of an algebra at the beginning of Section 6.2, I believe the only change required
is to replace "If a multiplication is defined" by
"If a bilinear multiplication is defined".

[fingolfin]

@lhsoicher thank you, sounds good to me. Now someone just needs to find time to add this. I'll add a "good first issue" here so that perhaps we can take care of it during the upcoming GAP days (if nobody does it before)

[fingolfin]

(oh I just see that "good first issue" was already there -- even better

[ThomasBreuer]

@lhsoicher Just for curiosity:
If precise definitions of vector spaces and algebras are regarded as necessary in the GAP Tutorial, why are they not requested also for groups, group actions, group homomorphisms, and algebra homomorphisms?

[lhsoicher]

@ThomasBreuer The problem with vector spaces and algebras in the GAP tutorial is that incomplete, and hence wrong, definitions are currently given. If you decide to define an object in the documentation then it should be a correct definition!