Add is_symmetric for permutation groups

[sylee957]

References to other Issues or PRs

Brief description of what is fixed or changed

I think that is_symmetric predicate is missing even if _is_sym cache is used in permutation groups.
I've used a naive test that the symmetric group is the largest permutation group that can be created from a base set of cardinality n, and have an order of n!.

Other comments

I think that permutation group is using custom cache for assumptions. But if there is a way to directly manipulate the global LRU cache or assumptions0 cache if some predicates are revealed during the computation, it may have to use those.

Release Notes

• combinatorics
  • Added a new predicate is_symmetric to test out if a permutation group is a symmetric group.
  • Added a new predicate is_alternating to test out if a permutation group is a alternating group.

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• combinatorics

  • Added a new predicate is_symmetric to test out if a permutation group is a symmetric group. (#17677 by @sylee957)

  • Added a new predicate is_alternating to test out if a permutation group is a alternating group. (#17677 by @sylee957)

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#### References to other Issues or PRs
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#### Brief description of what is fixed or changed

I think that `is_symmetric` predicate is missing even if `_is_sym` cache is used in permutation groups.
I've used a naive test that the symmetric group is the largest permutation group that can be created from a base set of cardinality `n`, and have an order of `n!`.

#### Other comments

I think that permutation group is using custom cache for assumptions. But if there is a way to directly manipulate the global LRU cache or assumptions0 cache if some predicates are revealed during the computation, it may have to use those.

#### Release Notes

<!-- Write the release notes for this release below. See
https://github.com/sympy/sympy/wiki/Writing-Release-Notes for more information
on how to write release notes. The bot will check your release notes
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<!-- BEGIN RELEASE NOTES -->
- combinatorics
  - Added a new predicate `is_symmetric` to test out if a permutation group is a symmetric group.
  - Added a new predicate `is_alternating` to test out if a permutation group is a alternating group.
<!-- END RELEASE NOTES -->

Update

The release notes on the wiki have been updated.

[jksuom]

Permutation.is_alt_sym is related. It is intended for large groups whose order is not easily found.

[sylee957]

One problem of using is_alt_sym for quicker test for True is that it treats symmetric group and alternating group as same.
I'm not sure how the algorithm behind _check_cycles_alt_sym can be improved to target only one predicate specifically. But I'll take a look.

[codecov]

Codecov Report

Merging #17677 into master will increase coverage by 0.003%.
The diff coverage is 61.176%.

@@             Coverage Diff              @@
##           master    #17677       +/-   ##
============================================
+ Coverage   74.68%   74.684%   +0.003%     
============================================
  Files         634       634               
  Lines      165598    165715      +117     
  Branches    38961     38977       +16     
============================================
+ Hits       123670    123764       +94     
- Misses      36442     36464       +22     
- Partials     5486      5487        +1

[sylee957]

Unfortunately, I don’t think that the jordan’s theorem can discriminate between an alternating group or a symmetric group, so there would not be able to modify the algorithm.

is_alternating may also need to be added for the reason.

[jksuom]

It is not hard to distinguish between Alt and Sym: If all generators are even, the group is Alt, otherwise it is Sym.

[sylee957]

I have some questions that if G.is_transistive() == False can also be a sufficient condition for is_alt == False and is_sym == False, and the cache can be updated.
I believe it is true from http://mathworld.wolfram.com/TransitiveGroup.html

I also think that the monte-carlo test can use None instead of False some uncertain false.
I would just want the direction to be more clear that

1. Add is_alternating and is_symmetric helpers.
2. Modify is_alt_sym to give None instead of some uncertain False.

[jksuom]

Looks good.