ClassifyFiniteSimpleGroup - Maple Help

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GroupTheory

 ClassifyFiniteSimpleGroup
 classify a finite simple group

 Calling Sequence ClassifyFiniteSimpleGroup( G ) ClassifyFiniteSimpleGroup( n )

Parameters

 G - a finite simple group n - a positive integer; the order of a finite simple group

Description

 • The ClassifyFiniteSimpleGroup( G ) command returns an object of type CFSG describing the classification of the finite simple group G as belonging to one of several families of finite simple groups as described below.
 • Since much of the classification depends solely on the order of G, you can also use a positive integer n that is the order of some finite simple group. In this case, if there are two possibilities (and there are at most two for any given order), an expression sequence of CFSG objects is returned describing the two possibilities.
 • The returned CFSG object c supports several methods that you can use to query the object for information about how the given group fits into the classification of finite simple groups. See CFSG for details on these methods, and for more information about the classification itself.
 • If the group G is not simple, or if the positive integer n is not the order of a finite simple group, then an exception is raised.
 • By default, the ClassifyFiniteSimpleGroup checks that the input group is simple. Since this can be expensive for a large group, if you know that your group is simple, you can avoid this check by passing the check = false option. This has no effect when the first argument is an integer n.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $\mathrm{ClassifyFiniteSimpleGroup}\left(\mathrm{Alt}\left(5\right)\right)$
 $⟨{\mathbf{\text{CFSG:}}}{\text{Alternating Group}}{{\mathbf{A}}}_{{5}}⟩$ (1)
 > $\mathrm{ClassifyFiniteSimpleGroup}\left(\mathrm{Alt}\left(8\right)\right)$
 $⟨{\mathbf{\text{CFSG:}}}{\text{Alternating Group}}{{\mathbf{A}}}_{{8}}⟩$ (2)
 > $\mathrm{ClassifyFiniteSimpleGroup}\left(\mathrm{Alt}\left(500\right)\right)$
 $⟨{\mathbf{\text{CFSG:}}}{\text{Alternating Group}}{{\mathbf{A}}}_{{500}}⟩$ (3)
 > $\mathrm{ClassifyFiniteSimpleGroup}\left(\mathrm{PSL}\left(3,4\right)\right)$
 $⟨{\mathbf{\text{CFSG:}}}{\text{Chevalley Group}}{{A}}_{{2}}{}\left({4}\right){=}{\mathrm{PSL}}{}\left({3}{,}{4}\right)⟩$ (4)
 > $\mathrm{ClassifyFiniteSimpleGroup}\left(\mathrm{CyclicGroup}\left(17\right)\right)$
 $⟨{\mathbf{\text{CFSG:}}}{\text{Cyclic Group}}{{C}}_{{17}}⟩$ (5)
 > $\mathrm{ClassifyFiniteSimpleGroup}\left(\mathrm{HeldGroup}\left(\right)\right)$
 $⟨{\mathbf{\text{CFSG:}}}{\text{Sporadic Group}}{\text{He}}⟩$ (6)
 > $G≔\mathrm{GraphTheory}:-\mathrm{AutomorphismGroup}\left(\mathrm{GraphTheory}:-\mathrm{SpecialGraphs}:-\mathrm{HoffmanSingletonGraph}\left(\right)\right)$
 ${G}{≔}{\mathrm{< a permutation group on 50 letters with 5 generators >}}$ (7)
 > $\mathrm{ClassifyFiniteSimpleGroup}\left(\mathrm{DerivedSubgroup}\left(G\right)\right)$
 $⟨{\mathbf{\text{CFSG:}}}{\text{Steinberg Group}}{}^{{2}}{A}_{{2}}{}\left({5}\right){=}{\mathrm{PSU}}{}\left({3}{,}{5}\right)⟩$ (8)
 > $\mathrm{ClassifyFiniteSimpleGroup}\left(\mathrm{OrthogonalGroup}\left("O10+\left(2\right)"\right)\right)$
 $⟨{\mathbf{\text{CFSG:}}}{\text{Chevalley Group}}{{\mathrm{D}}}_{{5}}{}\left({2}\right){=}{{\mathrm{PΩ}}}^{{+}}{}\left({10}{,}{2}\right)⟩$ (9)
 > $\mathrm{ClassifyFiniteSimpleGroup}\left(\mathrm{ithprime}\left(100\right)\right)$
 $⟨{\mathbf{\text{CFSG:}}}{\text{Cyclic Group}}{{C}}_{{541}}⟩$ (10)
 > $\mathrm{ClassifyFiniteSimpleGroup}\left({2}^{41}{3}^{13}{5}^{6}{7}^{2}\cdot 11\cdot 13\cdot 17\cdot 19\cdot 23\cdot 31\cdot 47\right)$
 $⟨{\mathbf{\text{CFSG:}}}{\text{Sporadic Group}}{\text{}}⟩$ (11)

The alternating group of degree $4$ is not simple, so the following command raises an exception.

 > $\mathrm{ClassifyFiniteSimpleGroup}\left(\mathrm{Alt}\left(4\right)\right)$

The check = false option can be useful in a situation like the following.

 > $G≔⟨⟨a,b⟩|⟨{a}^{2},{b}^{3},{\left(\mathrm{.}\left(a,b\right)\right)}^{5}=1⟩⟩$
 ${G}{≔}⟨{}{a}{,}{b}{}{\mid }{}{{a}}^{{2}}{,}{{b}}^{{3}}{,}{a}{}{b}{}{a}{}{b}{}{a}{}{b}{}{a}{}{b}{}{a}{}{b}{}⟩$ (12)
 > $\mathrm{ClassifyFiniteSimpleGroup}\left(G\right)$
 > $\mathrm{ClassifyFiniteSimpleGroup}\left(G,'\mathrm{check}'=\mathrm{false}\right)$
 $⟨{\mathbf{\text{CFSG:}}}{\text{Alternating Group}}{{\mathbf{A}}}_{{5}}⟩$ (13)

Alternatively since, in this case, the group is small, you could convert it to a permutation group, as follows.

 > $\mathrm{ClassifyFiniteSimpleGroup}\left(\mathrm{PermutationGroup}\left(G\right)\right)$
 $⟨{\mathbf{\text{CFSG:}}}{\text{Alternating Group}}{{\mathbf{A}}}_{{5}}⟩$ (14)

Compatibility

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