Given a group H and an element x belonging to a group K such that H and K are subgroups of the same symmetric group, this function returns the set of conjugates of x under the action of H. If H = K, the function returns the conjugacy class of x in H.
Given a group G, construct the conjugacy classes and the class map f for G. For any element x of G, f(x) will be the conjugacy class representative chosen by the Classes function. If the parameter Orbits is set true, the classes are computed as orbits of elements under conjugation and the class map is stored as a list of images of the elements of G (a list of length |G|). This option gives fast evaluation of the class map but is practical only in the case of very small groups. With Orbits := false, WeakLimit and StrongLimit are used to control the random classes algorithm (see function Classes).
WeakLimit: RngIntElt Default: 200
StrongLimit: RngIntElt Default: 500
Construct a set of representatives for the conjugacy classes of G. The classes are returned as a sequence of triples containing the element order, the class length and a representative element for the class. The parameters Reps and Al enable the user to select the algorithm that is to be used.
Reps: [ GrpMatElt ] Default:Reps := { Q}: Create the classes of G by assuming that Q is a sequence of class representatives for G. The orders and lengths of the classes will be computed and checked for consistency.
Al: MonStg Default:Al := "Action": Create the classes of G by computing the orbits of the set of elements of G under the action of conjugation. This option is only feasible for small groups.
Al := "Random": Construct the conjugacy classes of elements for a matrix group G using an algorithm that searches for representatives of all conjugacy of G by examining a random selection of group elements and their powers. The behaviour of this algorithm is controlled by two associated optional parameters WeakLimit and StrongLimit, whose values are positive integers n_1 and n_2, say. Before describing the effect of these parameters, some definitions are needed: A mapping f: G -> I is called a class invariant if f(g) = f(g^h) for all g, h in G. In matrix groups, the primary invariant factors are used where possible, or the characteristic or minimal polynomials otherwise. Two matrices g and h are said to be weakly conjugate with respect to the class invariant f if f(g) = f(h). By definition, conjugacy implies weak conjugacy, but the converse is false. The random algorithm first examines n_1 random elements and their powers, using a test for weak conjugacy. It then proceeds to examine a further n_2 random elements and their powers, using a test for ordinary conjugacy. The idea behind this strategy is that the algorithm should attempt to find as many classes as possible using the very cheap test for weak conjugacy, before employing the more expensive ordinary conjugacy test to recognize the remaining classes.
Default: The action algorithm will be used if |G| <= 2000. Otherwise the random algorithm will be applied with the limits given by the parameters WeakLimit and StrongLimit. If that fails to compute all the classes and |G| <= 100000, then the action algorithm will be used.
Given a group G for which the conjugacy classes are known and an element x of G, return the designated representative for the conjugacy class of G containing x.
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 returns a second value in the event that the elements are conjugate: an element k which conjugates g into h.
Given a group G and subgroups H and K belonging to G, return the value true if G and H 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.
The exponent of the group G.
The number of conjugacy classes of elements for the group G.
Given a group G, construct the power map for G. Suppose that the order of G is m and that G has r conjugacy classes. When the classes are determined by Magma, they are numbered from 1 to r. Let C be the set of class indices { 1, ..., r } and let P be the set of integers { 1, ..., m }. The power map f for G is the mapping, f : C x P -> C where the value of f(i, j) for i in C and j in P is the number of the class which contains x_i^j, where x_i is a representative of the i-th conjugacy class.
Given a group G, and a sequence Q of k distinct elements of G, one from each conjugacy class, use Q to define the classes attribute of G.[Next] [Prev] [Right] [Left] [Up] [Index] [Root]