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3 Related Algorithms
 3.1 Matrix groups
 3.2 Free groups

3 Related Algorithms

3.1 Matrix groups

3.1-1 ConjugacyGL2Z
‣ ConjugacyGL2Z( A, B )( function )

Decides whether a pair of 2x2 matrices in \(\mathrm{GL}_2(\mathbb{Z})\) are conjugate. If so returns the element \(C\) such that \(A=C^{-1}BC\).

3.1-2 CentralizerGL2Z
‣ CentralizerGL2Z( A, B )( function )

Given a matrix \(A\), computes a matrix \(C\) such that \(C_{\mathrm{GL}_2(\mathbb{Z})}(A) = \langle -C, C \rangle\) and an integer \(n\) such that \(C^n=A\).

3.1-3 MembershipSubgroupSL2Z
‣ MembershipSubgroupSL2Z( gens, A )( function )

Given a matrix \(A\), decides whether \(A\in\langle gens \rangle\) and if so computes a word \(w\) in the generators that represent \(A\).

3.2 Free groups

3.2-1 AreAutomorphicEquivalent
‣ AreAutomorphicEquivalent( F, u, v )( operation )

Decides whether u and v are automorphic equivalent in F.

3.2-2 IsCyclicalyReducedWord
‣ IsCyclicalyReducedWord( u )( property )

Returns: true or false

3.2-3 IsSubword
‣ IsSubword( u, v )( operation )

Returns whether the word v is a subword of u or not.

3.2-4 IsPrefix
‣ IsPrefix( u, v )( operation )

Returns whether the word v is a prefix of u or not.

3.2-5 IsSuffix
‣ IsSuffix( u, v )( operation )

Returns whether the word v is a suffix of u or not.

3.2-6 RootFreeGroup
‣ RootFreeGroup( v )( operation )

Returns a word \(w\) and an integer \(r\) such that \(w^r=v\) and \(r\) is maximal.

3.2-7 NielsenReducedSet
‣ NielsenReducedSet( V )( function )

Given a sequence of elements \(V\) of a free group, returns a Nielsen reduced set \(W\) that is equivalent to \(V\).

3.2-8 IsNielsenReducedSet
‣ IsNielsenReducedSet( V )( function )

Given a sequence of elements \(V\) of a free group, returns whether \(V\) is a Nielsen reduced set or not.

3.2-9 InFreeSubgroup
‣ InFreeSubgroup( V, w )( function )

Given a sequence of elements \(V\) of a free group and a word \(w\), returns whether \(w\) is in the subgroup generated by \(V\) or not.

3.2-10 IsSubgroupOfFreeSubgroup
‣ IsSubgroupOfFreeSubgroup( V, W )( function )

Given two sequence of elements \(V,W\) of a free group and a word \(w\), returns whether the subgroup generated by \(W\) is a subgroup of the group generated by \(V\) or not.

3.2-11 IsNormalSubgroupOfFreeSubgroup
‣ IsNormalSubgroupOfFreeSubgroup( V, W )( function )

Given two sequence of elements \(V,W\) of a free group and a word \(w\), returns whether the subgroup generated by \(W\) is a normal subgroup of the group generated by \(V\) or not.

3.2-12 AreEquivalent
‣ AreEquivalent( V, W )( function )

Given two sequence of elements \(V,W\) of a free group and a word \(w\), returns whether \(V\) and \(W\) are equivalent or not.

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