Add dicyclic_group, is_dicyclic_group and have quaternion_group and is_quaternion_group be aliases of those

[jamesnohilly]

This PR adds a group constructor for Dicyclic Groups (dicyclic_group) and changes the constructor for Quaternion Groups to be an alias of dicyclic_group.

This PR closes #1630

[thofma]

Suggested change

	where `n` is a power of 2 and `T` is in
	where `n` is a multiple of 4 and `T` is in

[lgoettgens]

I think "power of 2" is what we want here, as that is how (generalized) quaternion groups are defined. For the function that does the exact same construction for "multiple of 4", there is dicyclic_group below. In particular, this distinction in the docstring matches to how is_quaternion_group works

[thofma]

Hm, I think that the consensus in #1630 is that both should be the same. Here are some reasons why I think this is the case:

1. The definition with 2-powers is unnecessarily restrictive. There are plenty of references that construct quaternion groups of order 4n.
2. There is now already code out there that is using the most general definition (that the order is any number $4n$), because this is what people were being told in Dicyclic groups #1630
3. Since we don't do any quaternion_group(k) -> group of order 2^k or 2^(k+2) shenanigans, there is no reason to restrict to 2-powers.

[fingolfin]

Yeah I decided to just treat both commands as exact equals, any artificial restriction benefits nobody but can cause frustration

[fingolfin]

Note that e.g. Wikipedia allows "generalized quaternion groups" to have order $4k$.

[lgoettgens]

I think "power of 2" is what we want here, as that is how (generalized) quaternion groups are defined. For the function that does the exact same construction for "multiple of 4", there is dicyclic_group below. In particular, this distinction in the docstring matches to how is_quaternion_group works

[codecov]

Codecov Report

Attention: Patch coverage is 91.89189% with 6 lines in your changes missing coverage. Please review.

Project coverage is 84.77%. Comparing base (4bccbc1) to head (ca99ed5).
Report is 64 commits behind head on master.

Files with missing lines	Patch %	Lines
src/Groups/group_constructors.jl	91.89%	6 Missing ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##           master    #4661      +/-   ##
==========================================
+ Coverage   84.38%   84.77%   +0.39%     
==========================================
  Files         673      681       +8     
  Lines       90284    91838    +1554     
==========================================
+ Hits        76185    77859    +1674     
+ Misses      14099    13979     -120     

Files with missing lines	Coverage Δ
src/Groups/group_constructors.jl	93.65% <91.89%> (+0.27%)	⬆️

... and 96 files with indirect coverage changes

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[thofma]

Very nice, thanks. But some group theory person should have a look too.

[fingolfin]

Adjust this to fully mirror the quaternion_group docstring (including a @ref link in the other direction).

... or... perhaps we should instead really merge the two, like GAP does: show a single docstring that starts with

"""
    dicyclic_group(::Type{T} = PcGroup, n::IntegerUnion)
    quaternion_group(::Type{T} = PcGroup, n::IntegerUnion)

...

Then attach that to, say, quaternion_group, and turn dicyclic_group into a "true" alias by doing const dicyclic_group = quaternion_group (I think it'll then get the same docstring). Merge is_quaternion_group and is_dicyclic_group` similarly.

In the examples use only one (quaternion).

Then instead of This is an alias of dicyclic_group. write something more like this:

!!! note
    For historical reasons and backwards compatibility, `dicyclic_group` is an alias
    of `quaternion_group`. The two functions are fully identical. We recommend always
    using `quaternion_group`.

and then similar for the is_* function.

Thoughts @ThomasBreuer @thofma ?

[thofma]

If it is just an alias, as a user I would not quite understand why you provide an alias that I am happy to find (because "dicyclic group" is the right name in my domain) and use, but then tell me not to use it.

But I don't have a strong opinion on this.

[ThomasBreuer]

My understanding of the discussion from issue #1630 is that it should be the other way round, that is, we recommend always using dicyclic_group.

And yes, I think it makes sense to have just one docstring for both names.

[fingolfin]

I am fine with not making a recommendation, my main concern is that I think it'd be nice to mention why we have two names.

[ThomasBreuer]

I am getting confused.
GAP does the following:

gap> G:= DicyclicGroup(24);;
gap> IsQuaternionGroup(G);
false
gap> G:= DicyclicGroup(32);;
gap> IsQuaternionGroup(G);
true

That is, both QuaternionGroup and DicyclicGroup can be used to create the groups in question,
but IsQuaternionGroup regards only the dicyclic groups of 2-power order as (generalized) quaternion groups.
In fact, GAP seems to have no function that checks whether a group is dicyclic.

[ThomasBreuer]

As far as I see, the GAP function DoComputeGeneralisedQuaternionGenerators (which is called by IsQuaternionGroup) would work as an is_dicyclic check if one just omits the check for 2-power order.

[ThomasBreuer]

How shall we proceed?
Meanwhile gap-system/gap/pull/5966 got merged, do we want to wait until Oscar can use it, or do we want to add Oscar code that provides the same functionality?

[jamesnohilly]

I have now merged the docstrings of dicyclic_group and quaternion_group, with main preference on dicyclic_group as in #1630.

As for the is_{dicyclic,quaternion}_group issue, we could wait for a new GAP.jl release to use these new methods, however I am unsure. Adding this functionality to OSCAR and then eventually replacing is an option aswell.

[thofma]

Didn't we decide a while ago that dicyclic_group === quaternion_group and is_dicyclic_group === is_quaternion_group (#1630 and #4661 (comment))? I think now having

julia> is_quaternion_group(quaternion_group(12))
false

is not optimal.

[ThomasBreuer]

Concerning @thofma's latest comment:
Yes, I think the solution is

• to make also is_dicyclic_group and is_quaternion_group synonyms, in the same way as dicyclic_group and quaternion_group are,
• to merge the docstrings of is_dicyclic_group and is_quaternion_group into one docstring, and
• to change the N // 2 to div(N, 2).

(The current code wants is_dicyclic_group and is_quaternion_group to behave analogous to the corresponding functions in GAP, but the situation in GAP is different, since GAP.Globals.GeneralizedQuaternionGroup prints a warning if the argument is not a 2-power, and GAP.Globals.IsGeneralizedQuaternionGroup returns false if the order of the argument is not a 2-power. In gap-system/gap/pull/5966, I had just added the missing function GAP.Globals.IsDicyclicGroup, without changing the -perhaps inconsistent- behaviour of GAP.Globals.IsGeneralizedQuaternionGroup and GAP.Globals.GeneralizedQuaternionGroup.)

[jamesnohilly]

Regarding this issue, I believe I became confused regarding GAP's functionality vs. OSCAR functionality throughout the reimplementation of is_dicyclic_group. However, I will implement @ThomasBreuer's solution in the evening and clean up this issue.

[ThomasBreuer]

Looks good. Thanks.

[fingolfin]

Just to add another data point that convinces me we are doing the right thing merging the two notations: Keith Conrad also considers them as synonyms in https://kconrad.math.uconn.edu/blurbs/grouptheory/genquat.pdf