[Next][Prev] [Right] [Left] [Up] [Index] [Root]

Standard Subgroup Constructions

Some functions described in this section may not exist or may have restrictions for some categories of groups. Details can be found in the chapters on the individual categories.

Subsections
H ^ g : GrpFin, GrpFinElt -> GrpFin
Conjugate(H, g) : GrpFin, GrpFinElt -> GrpFin
Construct the conjugate g - 1Hg of the group H by the element g. The group H and the element g must belong to the same generic group.
H meet K : GrpFin, GrpFin -> GrpFin
Given groups H and K which belong to the same symmetric group, construct the intersection of H and K.
CommutatorSubgroup(G, H, K) : GrpFin, GrpFin, GrpFin -> GrpFin
CommutatorSubgroup(H, K) : GrpFin, GrpFin -> GrpFin
Given groups H and K, both subgroups of the group G, construct the commutator subgroup of H and K in the group G. If K is a subgroup of H, then the group G may be omitted.
Centralizer(G, g) : GrpFin, GrpFinElt -> GrpFin
Centraliser(G, g) : GrpFin, GrpFinElt -> GrpFin
Construct the centralizer of the element g in the group G.
Centralizer(G, H) : GrpFin, GrpFin -> GrpFin
Centraliser(G, H) : GrpFin, GrpFin -> GrpFin
Construct the centralizer of the group H in the group G.
Core(G, H) : GrpFin, GrpFin -> GrpFin
Given a subgroup H of the group G, construct the maximal normal subgroup of G that is contained in the subgroup H.
H ^ G : GrpFin, GrpFin -> GrpFin
NormalClosure(G, H) : GrpFin, GrpFin -> GrpFin
Given a subgroup H of the group G, construct the normal closure of H in G.
Normalizer(G, H) : GrpFin, GrpFin -> GrpFin
Normaliser(G, H) : GrpFin, GrpFin -> GrpFin
Given a subgroup H of the group G, construct the normalizer of H in G.
pCore(G, p) : GrpFin, RngIntElt -> GrpFin
Given a group G and a prime p dividing the order of G, construct the maximal normal p-subgroup of G.
SylowSubgroup(G, p) : GrpFin, RngIntElt -> GrpFin
Sylow(G, p) : GrpFin, RngIntElt -> GrpFin
Given a group G and a prime p, construct a Sylow p-subgroup of G.

Abstract Group Predicates

Some functions described in this section may not exist or may have restrictions for some categories of groups. Details can be found in the chapters on the individual categories.

IsAbelian(G) : GrpFin -> BoolElt
Returns true if the group G is abelian, false otherwise.
IsCyclic(G) : GrpFin -> BoolElt
Returns true if the group G is cyclic, false otherwise.
IsElementaryAbelian(G) : GrpFin -> BoolElt
Returns true if the group G is elementary abelian, false otherwise.
IsCentral(G, H) : GrpFin -> BoolElt
Return true if the subgroup H of the group G lies in the centre of G, false otherwise.
IsConjugate(G, g, h) : GrpFin, GrpFinElt, GrpFinElt -> BoolElt, GrpFinElt
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) : GrpFin, GrpFin, GrpFin -> BoolElt, GrpFinElt
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 if the subgroups are conjugate: an element z which conjugates H into K.
IsExtraSpecial(G) : GrpFin -> BoolElt
Given a group G is a p-group G, return true if G is extra-special, false otherwise.
IsMaximal(G, H) : GrpFin, GrpFin -> BoolElt
Returns 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 one hundred thousand.
IsNilpotent(G) : GrpFin -> BoolElt
Return true if the group G is nilpotent, false otherwise.
IsNormal(G, H) : GrpFin, GrpFin -> BoolElt
Return true if the subgroup H of the group G is a normal subgroup of G, false otherwise.
IsPerfect(G) : GrpFin -> BoolElt
Return true if the group G is perfect, false otherwise.
IsSelfNormalizing(G, H) : GrpFin, GrpFin -> BoolElt
IsSelfNormalising(G, H) : GrpFin, GrpFin -> BoolElt
Return true if the subgroup H of the group G is self-normalizing in G, false otherwise.
IsSimple(G) : GrpFin -> BoolElt
Return true if the group G is simple, false otherwise.
IsSoluble(G) : GrpFin -> BoolElt
IsSolvable(G) : GrpFin -> BoolElt
Return true if the group G is soluble, false otherwise.
IsSpecial(G) : GrpFin -> BoolElt
Given a p-group G, return true if G is special, false otherwise.
IsSubnormal(G, H) : GrpFin, GrpFin -> BoolElt
Return true if the subgroup H of the group G is subnormal in G, false otherwise.
IsTrivial(G) : Grp -> BoolElt
Return true if G is trivial, false otherwise.
 [Next][Prev] [Right] [Left] [Up] [Index] [Root]

Version: V2.19 of Mon Dec 17 14:40:36 EST 2012