construct a Hall subgroup of a finite soluble group
HallSubgroup( pi, G )
a list or set of primes
a soluble permutation group
Let G be a finite group, and let pi be a set of (positive, rational) primes. A Hall pi-subgroup of G is a maximal pi-subgroup of G where, by a pi-subgroup, we mean a subgroup whose order is a pi-number (one whose prime divisors all belong to pi). Equivalently, a subgroup H of a finite group G is a Hall-subgroup if its order and index are relatively prime.
If pi consists of a single prime number p, then a Hall pi-subgroup of G is just a Sylow p-subgroup of G.
A finite group G is soluble if, and only if, for each set pi of primes, G has a Hall pi-subgroup. Moreover, any two Hall pi-subgroups of G are conjugate in G, and every pi-subgroup of G is contained in a Hall pisubgroup.
A finite insoluble group may, or may not, have Hall subgroups.
The HallSubgroup( pi, G ) command constructs a Hall pi-subgroup of a finite soluble group G. The group G must be an instance of a permutation group. Apart from a handful of exceptions, the permutation group G must be soluble; otherwise, an exception is raised.
G ≔ DihedralGroup⁡30
H ≔ HallSubgroup⁡2,5,G
Hall subgroups can only be computed for soluble groups, in general, so the following example cause an exception to be raised.
Error, (in GroupTheory:-HallSubgroup) group must be soluble
However, for certain special cases, a Hall subgroup is returned without exception.
The GroupTheory[HallSubgroup] command was introduced in Maple 17.
For more information on Maple 17 changes, see Updates in Maple 17.
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