‣ 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 frac1n.
‣ PrimeCommDegreeGroupFind( prime ) | ( function ) |
The PrimeCommDegreeGroupFind function receives a prime number p and returns a group G with commutativity degree frac1p, 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 frac1p_i.
‣ NilpotentCommDegreeGroupFind( number ) | ( function ) |
Similar to the PrimeCommDegreeGroupFind, but returning a nilpotent group with commutativity degree less than frac1n given an integer number n.
‣ NilpotentPrimeCommDegreeGroupFind( prime ) | ( function ) |
Similar to the PrimeCommDegreeGroupFind, but returning a nilpotent group with commutativity degree less than frac1p given a prime number p.
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