GAP (HAP) - Chapter 19: Words in free ZG-modules

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19 Words in free ZG-modules

### 19 Words in free ZG-modules

`AddFreeWords(v,w)` Inputs two words v,w in a free ZG-module and returns their sum v+w. If the characteristic of Z is greater than 0 then the next function might be more efficient.
`AddFreeWordsModP(v,w,p)` Inputs two words v,w in a free ZG-module and the characteristic p of Z. It returns the sum v+w. If p=0 the previous function might be fractionally quicker.
`AlgebraicReduction(w)` `AlgebraicReduction(w,p)` Inputs a word w in a free ZG-module and returns a reduced version of the word in which all pairs of mutually inverse letters have been cancelled. The reduction is performed in a free abelian group unless the characteristic p of Z is entered.
`Multiply Word(n,w)` Inputs a word w and integer n. It returns the scalar multiple n⋅ w.
`Negate([i,j])` Inputs a pair [i,j] of integers and returns [-i,j].
`NegateWord(w)` Inputs a word w in a free ZG-module and returns the negated word -w.
`PrintZGword(w,elts)` Inputs a word w in a free ZG-module and a (possibly partial but sufficient) listing elts of the elements of G. The function prints the word w to the screen in the form r_1E_1 + ... + r_nE_n where r_i are elements in the group ring ZG, and E_i denotes the i-th free generator of the module.
`TietzeReduction(S,w)` Inputs a set S of words in a free ZG-module, and a word w in the module. The function returns a word w' such that {S,w'} generates the same abelian group as {S,w}. The word w' is possibly shorter (and certainly no longer) than w. This function needs to be improved!
`ResolutionBoundaryOfWord(R,n,w)` Inputs a resolution R, a positive integer n and a list w representing a word in the free module R_n. It returns the image of w under the n-th boundary homomorphism.

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