Although the conjugacy classes of an abelian group are trivial, the standard class functions are provided for completeness.

Class(H, g) : GrpAb, GrpAbElt -> { GrpAbElt }

Conjugates(H, g) : GrpAb, GrpAbElt -> { GrpAbElt }

g ^ H : GrpAbElt, GrpAb -> { GrpAbElt }

Given a group H and an element g belonging to a group K such that H and K are subgroups of some covering group, this function returns the set of conjugates of g under the action of H. If H = K, the function returns the conjugacy class of g in H.

Classes(G) : GrpAb -> [ <RngIntElt, RngIntElt, GrpAbElt> ]

Construct a set of representatives for the conjugacy classes of G. The classes are returned as a sequence of tuples containing the class length, the order of the elements in the class and a representative element for the class.

The class map M: G -> {1, ..., n} for the group G, where n is the number of conjugacy classes of G.

The class matrix corresponding to the i-conjugacy class of elements for the group G.

The designated representative for the conjugacy class of G containing x (relative to existing conjugacy classes).

Given a group G and elements g and h belonging to G, return the value true if g and h are conjugate in G. The function also returns a second value in the event that the elements are conjugate: an element z which conjugates g into h.

IsConjugate(G, H, K) : GrpAb, GrpAb, GrpAb -> BoolElt, GrpAbElt

Given a group G and subgroups H and K belonging to G, return the value true if H and K are conjugate in G. The function returns a second value in the event that the subgroups are conjugate: an element z which conjugates H into K.

Nclasses(G) : GrpAb -> RngIntElt

The number of conjugacy classes of elements of the group G.

The power map M associated with the conjugacy classes of G. M describes where the elements of the conjugacy classes of G move under powers. That is, M(c, n) = < c, n > @M returns the class number where class c moves under the power n. The value of c must be in the range [1 ... Nclasses(G)].

M: {1 ... n} x Z -> {1 ... n}