Subsections

• Permutation Group Predicates
• Abstract Group Predicates

True if the permutation group G defined as acting on X is the alternating group Alt(X).

True if the permutation group G defined as acting on X is the symmetric group Sym(X).

True if the permutation group G defined as acting on X contains the alternating group Alt(X).

Given a permutation group defined as acting on X, return true if G acts primitively on X, false otherwise.

Given a permutation group G defined as acting on a set X, return true if G acts regularly on X (i.e. G acts transitively on X, and the stabilizer in G of any point in X is the identity).

Given a permutation group G and a set S containing a union of orbits of G, return true if G acts semiregularly on S, false otherwise. If S is omitted, then S is taken to be the natural G-set X of G. (A group G is said to act semiregularly on a set S if G acts regularly on each G-orbit contained in S.)

Given a permutation group G defined as acting on X, return true if G acts transitively on X, false otherwise.

Given a permutation group G defined as acting on X, and a non-negative integer k, return true if G acts sharply k-transitively on X, false otherwise.

True if the permutation group G is a Frobenius group with respect to its natural action, false otherwise. (A group G defined as acting on X is Frobenius if it acts transitively but non-regularly on X and if the pointwise stabilizer of any two distinct points of X is the trivial group.)

True if the group G is abelian, false otherwise.

True if the group G is cyclic, false otherwise.

True if the group G is elementary abelian, false otherwise.

True if the subgroup H of the group G lies in the centre of G, false otherwise.

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 if the elements are conjugate: an element k which conjugates g into h.

IsConjugate(G, H, K) : GrpPerm, GrpPerm, GrpPerm -> BoolElt, GrpPermElt

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 if the subgroups are conjugate: an element z which conjugates H into K.

Given a group G is a p-group G, return true if G is extra-special, false otherwise.

True if the subgroup H of the group G is a maximal subgroup of G. This function is evaluated by constructing the permutation representation of G on the cosets of H and testing this representation for primitivity. For this reason, the use of IsMaximal should be avoided if the index of H in G exceeds a few thousand.

True if the group G is nilpotent, false otherwise.

True if the subgroup H of the group G is a normal subgroup of G, false otherwise.

True if the group G is perfect, false otherwise.

IsSelfNormalising(G, H) : GrpPerm, GrpPerm -> BoolElt

True if the subgroup H of the group G is self-normalizing in G, false otherwise.

True if the group G is simple, false otherwise.

IsSolvable(G) : GrpPerm -> BoolElt

True if the group G is soluble, false otherwise.

Given a p-group G, return true if G is special, false otherwise.

True if the subgroup H of the group G is subnormal in G, false otherwise.