Deliverables

• Implementation of the collection algorithm
• Implement a method to compute coset representatives of a given CosetTable.(Might be useful in the computation of polycyclic series).
• Implement Elementatry methods like collected_word, compute_pcgs, minimal_uncollected_subword, compute_relative_orders etc…
• Make final changes to the modifed-Todd-Coxeter PR.
• Create a PR for subgroup_presentation methods.
• Maximal Abelian quotient

Algorithms

• Collection Algorithm: All the minimal uncollected subwords of the given word are computed and the every minimal uncollected subword in the given word is replaced by the the corresponding power relation. This is done till all the uncollected sub-words are eleminated from the given word. A detailed implementation of the algoeithm is mentioned in the section 8.1.3 of the Handbook.
• Coset Representaive: A simple backtracking algo where we first start at the given coset and find a gamma = C.table[coset][A.dict[x]] such that gamma < coset and return coset_rep(gamma)*x**-1
• Injectove maps from subgroup to parent group: This is basically computing an injective map for a subgroup defined o the new generators after the reidemester_presentation is called
• Computing the abelian-subgroup-quotient (Rewritten PR): The presentation of the quotient G/H is given by adding the generators of G to the relators of H, they are defined on the same free-group.
• Maximal abelian quotient: Largest abelian quotient will be the subgroup quotient of the parent group with the commutatot subgroup.

PRs

• Polycyclic PR.
• Injective homomorphism PR.
• Quotient methods PR.

Week-11:

• Conversion of the word with more than one elements in the polycylic presentation to the elemetns of another FreeGroup similar to the coin change problem
• Implementation of the Epimorphism quotient methods.
• Implement methods to compute the pc-presentation and the power realtions.
• To-Do: Add tests and documentation.

  References

• [1] Derek F.Holt, Bettina Fick, Eamonn A.O’Brian. Handbook of Computational Group Theory.
• [2] Article on Polycyclic groups.
• [3] GAP - Polycyclic package.
• [4] Groupprops