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.
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.
Given groups H and K which belong to the same symmetric group, construct the intersection of H and K.
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.
Construct the centralizer of the element g in the group G.
Construct the centralizer of the group H in the group G.
Given a subgroup H of the group G, construct the maximal normal subgroup of G that is contained in the subgroup H.
Given a subgroup H of the group G, construct the normal closure of H in G.
Given a subgroup H of the group G, construct the normalizer of H in G.
Given a group G and a prime p dividing the order of G, construct the maximal normal p-subgroup of G.
Given a group G and a prime p, construct a Sylow p-subgroup of G.
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.
Returns true if the group G is abelian, false otherwise.
Returns true if the group G is cyclic, false otherwise.
Returns true if the group G is elementary abelian, false otherwise.
Return 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.
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.
Given a group G is a p-group G, return true if G is extra-special, false otherwise.
p: RngIntElt Default: 0
Test if G is a finite hyperelementary (=quasi-elementary) group, that is a semi-direct product G=C:P with C cyclic and P a p-group of order coprime to C. If yes, returns true, C, P. Otherwise returns false. The prime number p can be forced with an optional parameter p (0 by default).
> ok,C,P:=IsHyperelementary(Sym(3)); // S3=C3:C2 is hyperelementary > ok,GroupName(C),GroupName(P); true C3 C2 > ok,C,P:=IsHyperelementary(Sylow(Sym(6),2)); // So is any p-group > ok,GroupName(C),GroupName(P); true C1 C2*D4 > IsHyperelementary(Sym(4)); // S4 is not hyperelementary false
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.
Return true if the group G is nilpotent, false otherwise.
Return true if the subgroup H of the group G is a normal subgroup of G, false otherwise.
Return true if the group G is perfect, false otherwise.
Returns true if the group G is a Q-group (all characters are rational-valued).
> IsQGroup(Sym(4)); // Symmetric groups S_n are Q-groups true > IsQGroup(Alt(4)); // and alternating groups A_n are not (for n>2) false
Return true if the subgroup H of the group G is self-normalizing in G, false otherwise.
Return true if the group G is simple, false otherwise.
Return true if the group G is soluble, false otherwise.
Given a p-group G, return true if G is special, false otherwise.
Return true if the subgroup H of the group G is subnormal in G, false otherwise.
Return true if G is trivial, false otherwise.