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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.
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.
Given groups H and K which belong to the same symmetric
group, construct the intersection of H and K.
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.
Centraliser(G, g) : GrpFin, GrpFinElt -> GrpFin
Construct the centralizer of the element g in the group G.
Centraliser(G, H) : GrpFin, GrpFin -> GrpFin
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.
NormalClosure(G, H) : GrpFin, GrpFin -> GrpFin
Given a subgroup H of the group G, construct
the normal closure of H in G.
Normaliser(G, H) : GrpFin, GrpFin -> GrpFin
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.
Sylow(G, p) : GrpFin, RngIntElt -> GrpFin
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
IsSelfNormalising(G, H) : GrpFin, GrpFin -> BoolElt
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.
IsSolvable(G) : GrpFin -> BoolElt
Return true if the group G is soluble, false otherwise.
Given a p-group G, return true if G is special, false
otherwise.
Is the finite group G a Frobenius group as an abstract group? Equivalently,
does there exist a faithful permutation representation of G for which the
image is a Frobenius group as a permutation group? If so, then return true,
a Frobenius kernel, and a corresponding Frobenius complement; otherwise
return false.
Return true if the subgroup H of the group G is subnormal in G,
false otherwise.
Return true if G is trivial, false otherwise.
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