Group Theory
Expanded Small Groups Database and
Database Search Commands
The database of small groups included in the GroupTheory package has been expanded
to include all groups of order less than
. Previously, only groups up to order
were
included.
Searching of the small groups database and the database of transitive groups are
implemented in the new SearchSmallGroups and SearchTransitiveGroups commands,
respectively.
>
Find the non-Abelian simple groups in the database; a sequence of small group IDs is
returned.
>
(1)
Find groups in the database of order less than
with Sylow -subgroup of order , and
an unique Sylow -subgroup, and output them as finitely presented groups.
>
(2)
Find how many non-nilpotent groups in the database have derived subgroup isomorphic
to the quaternion group of order .
>
101
(3)
Output the transitive group IDs of the regular permutation groups of degree .
>
(4)
>
(5)
>
6
(6)
true
(7)
>
New Cayley Graph Visualization
A new command for computing and visualizing Cayley graphs of small groups is
included in the GroupTheory package.
>
>
(8)
>
>
>
>
>
(9)
>
>
Other New Commands
The following additional commands are new in this release: ComplexProduct,
ElementOrder, Exponent, FreeGroup, IsCyclic,
A complex in a group is just a subset of a group. The ComplexProduct command
computes the product of two complexes and in a group , which is defined to be the
set of products of the form
, for in and in .
>
(10)
(10)
>
>
>
>
>
>
>
(11)
>
>
>
(12)
The new ElementOrder command computes the order of an element of a finite group,
represented as either a permutation group or a Cayley table group.
>
>
(13)
>
(14)
>
(15)
>
2
(16)
The Exponent command computes the exponent of a finite group represented as either
a permutation group or a Cayley table group.
>
(17)
(17)
>
3
(18)
6
(19)
2
(20)
>
>
To construct a free group as a finitely presented group, use the new FreeGroup
command.
>
(21)
>
true
(22)
false
(23)
>
The IsCyclic command attempts to determine whether a group is cyclic.
>
false
(24)
>
(25)
>
true
(26)
>
(27)
>
false
(28)