‣ CommGraph( group ) | ( function ) |
This function receives a group as an input then utilizes the Grape package to implement a graph \((V,E)\) applying the following rule:
The vertices of this graph are the elements of the inputted group \(G\) and two vertices are connected if the elements commute
‣ DeepCommGraph( group ) | ( function ) |
This function receives a group as an input then utilizes the Grape package to implement a graph \((V,E)\) applying the following rule:
Two elements of \(G\) are joined in the deep commuting graph if and only if their preimages in every central extension of \(G\) (that is, every group \(H\) with a central subgroup \(Z\) such that \(H/Z \cong G\)) commute.
‣ CommDegree( group ) | ( function ) |
The commutativity degree of a group is defined as the probability that two elements of the group commutes.
The CommDegree function receives a group as an input then outputs a number \(\text{CommDegree}(G) \in (\frac{5}{8}, 1)\) that represents this probability. If the group is abelian, then the commutative degree is 1. Otherwise it is always less than \(\frac{5}{8}\) as a consequence of the \(\frac{5}{8}\) lower bound theorem.
‣ MaxAbelianSubgroup( group ) | ( function ) |
This function receives a group as an input then utilizes the Grape package function MaximumClique(graph) to return the group generated by a maximum clique of the group's commutative graph.
‣ CommDegreeGroupFind( group ) | ( function ) |
This function receives an integer number \(n\) and then outputs a group \(G\) with commutativity degree \(\frac{1}{n}\).
‣ PrimeCommDegreeGroupFind( prime ) | ( function ) |
The PrimeCommDegreeGroupFind function receives a prime number \(p\) and returns a group \(G\) with commutativity degree \(\frac{1}{p}\), the CommDegreeGroupFind function uses this function taking the prime factorization of the number \(n\) and taking the direct product of the groups with commmutativity degree equal to \(\frac{1}{p_i}\).
‣ NilpotentCommDegreeGroupFind( number ) | ( function ) |
Similar to the PrimeCommDegreeGroupFind, but returning a nilpotent group with commutativity degree less than \(\frac{1}{n}\) given an integer number \(n\).
‣ NilpotentPrimeCommDegreeGroupFind( prime ) | ( function ) |
Similar to the PrimeCommDegreeGroupFind, but returning a nilpotent group with commutativity degree less than \(\frac{1}{p}\) given a prime number \(p\).
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