CFSG - Maple Help

GroupTheory

 CFSG
 object classifying finite simple groups

Description

 • A CFSG object is a Maple object that classifies finite simple groups and is returned by the ClassifyFiniteSimpleGroup command. Its purpose is to give a "name" to a finite simple group according to the classification of finite simple groups.

Methods

 • Objects of the CFSG support a number of methods, as follows.
 • The Family( c ) command returns a string naming the family to which the group belongs.
 • In some cases, the Subfamily( c ) command returns a string indicating a sub-family of some families of simple groups. This is the case for some of the sporadic finite simple groups, and for those finite simple groups of Lie type. In other cases, the Subfamily( c ) command just returns the value undefined, indicating that there is no sub-family involved.
 • If the CFSG object c denotes a classical group, then the ClassicalSubfamily( c ) command returns a string indicating the type of classical group. For other groups of Lie type (or simple groups not of Lie type), it returns the value undefined.
 • The Parameters( c ) command returns a list of parameters associated with the group, if any, or the value undefined in case there are none. In most cases, there are either one or two positive integer parameters describing the group within a family or sub-family of simple groups. For example, the parameter associated with an alternating group denotes the degree.
 • The GroupOrder( c ) command returns the order of the group.
 • The commands IsCyclic( c ), IsAlternating( c ), IsLieType( c ) and IsSporadic( c ) return true or false according to whether the CFSG object c represents a classifier belonging to the corresponding family. In addition, for CFSG objects representing simple groups of Lie type, the IsClassical( c ) command returns true if the classifier belongs to a family of classical groups.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $c≔\mathrm{ClassifyFiniteSimpleGroup}\left(\mathrm{CyclicGroup}\left(541\right)\right)$
 ${c}{≔}⟨{\mathbf{\text{CFSG:}}}{\text{Cyclic Group}}{{C}}_{{541}}⟩$ (1)
 > $\mathrm{type}\left(c,':-\mathrm{CFSG}'\right)$
 ${\mathrm{true}}$ (2)
 > $\mathrm{Family}\left(c\right)$
 ${"cyclic"}$ (3)

The Subfamily is not defined for simple cyclic groups.

 > $\mathrm{Subfamily}\left(c\right)$
 ${\mathrm{undefined}}$ (4)
 > $\mathrm{Parameters}\left(c\right)$
 $\left[{541}\right]$ (5)
 > $c≔\mathrm{ClassifyFiniteSimpleGroup}\left(\mathrm{Alt}\left(22\right)\right)$
 ${c}{≔}⟨{\mathbf{\text{CFSG:}}}{\text{Alternating Group}}{{\mathbf{A}}}_{{22}}⟩$ (6)
 > $\mathrm{type}\left(c,':-\mathrm{CFSG}'\right)$
 ${\mathrm{true}}$ (7)
 > $\mathrm{Family}\left(c\right)$
 ${"alternating"}$ (8)
 > $\mathrm{Subfamily}\left(c\right)$
 ${\mathrm{undefined}}$ (9)
 > $\mathrm{Parameters}\left(c\right)$
 $\left[{22}\right]$ (10)
 > $\mathrm{Degree}\left(c\right)$
 ${22}$ (11)
 > $c≔\mathrm{ClassifyFiniteSimpleGroup}\left(\mathrm{PSL}\left(4,3\right)\right)$
 ${c}{≔}⟨{\mathbf{\text{CFSG:}}}{\text{Chevalley Group}}{{A}}_{{3}}{}\left({3}\right){=}{\mathrm{PSL}}{}\left({4}{,}{3}\right)⟩$ (12)
 > $\mathrm{Family}\left(c\right)$
 ${"Lie"}$ (13)
 > $\mathrm{Subfamily}\left(c\right)$
 ${"A"}$ (14)
 > $\mathrm{ClassicalSubfamily}\left(c\right)$
 ${"PSL"}$ (15)
 > $\mathrm{Parameters}\left(c\right)$
 $\left[{3}{,}{3}\right]$ (16)
 > $\mathrm{LieRank}\left(c\right)$
 ${3}$ (17)
 > $\mathrm{Dimension}\left(c\right)$
 ${4}$ (18)
 > $c≔\mathrm{ClassifyFiniteSimpleGroup}\left(\mathrm{RudvalisGroup}\left(\right)\right)$
 ${c}{≔}⟨{\mathbf{\text{CFSG:}}}{\text{Sporadic Group}}{\text{Ru}}⟩$ (19)
 > $\mathrm{Family}\left(c\right)$
 ${"sporadic"}$ (20)
 > $\mathrm{Subfamily}\left(c\right)$
 ${"Rudvalis"}$ (21)
 > $\mathrm{Parameters}\left(c\right)$
 ${\mathrm{undefined}}$ (22)
 > $c≔\mathrm{ClassifyFiniteSimpleGroup}\left(\mathrm{JankoGroup}\left(2\right)\right)$
 ${c}{≔}⟨{\mathbf{\text{CFSG:}}}{\text{Sporadic Group}}{{J}}_{{2}}⟩$ (23)
 > $\mathrm{Family}\left(c\right)$
 ${"sporadic"}$ (24)
 > $\mathrm{Subfamily}\left(c\right)$
 ${"Janko"}$ (25)
 > $\mathrm{Parameters}\left(c\right)$
 $\left[{2}\right]$ (26)

Compatibility

 • The GroupTheory[CFSG] command was introduced in Maple 2020.