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GroupTheory

 NumSimpleGroups
 return the number of simple groups of a given finite order

 Calling Sequence NumSimpleGroups( n )

Parameters

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

Description

 • For a positive integer n, the NumSimpleGroups( n ) command returns the number of simple groups of order n.
 • For each prime integer n, the number of simple groups of order n is equal to $1$. By the Feit-Thompson Theorem, if n is an odd composite integer, there are no simple groups of order n. The Artin-Tits theorem asserts that the number of simple groups of any given finite order is at most equal to $2$. Therefore, the value returned by NumSimpleGroups( n ), for any positive integer n, is one of $0$, $1$ or $2$.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $\mathrm{NumSimpleGroups}\left(1\right)$
 ${0}$ (1)
 > $\mathrm{NumSimpleGroups}\left(13\right)$
 ${1}$ (2)
 > $\mathrm{NumSimpleGroups}\left(15\right)$
 ${0}$ (3)
 > $\mathrm{NumSimpleGroups}\left(360\right)$
 ${1}$ (4)
 > $\mathrm{NumSimpleGroups}\left(20160\right)$
 ${2}$ (5)

Compatibility

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