Alternating group:A8

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VIEW FACTS ABOUT THIS GROUP: All factsGroup property satisfactionsSubgroup property satisfactionsGroup property dissatisfactionsSubgroup property satisfactions
VIEW DEFINITIONS USING THIS AS EXAMPLE: All terms |Group properties |Subgroup properties|
VIEW FACTS USING THIS AS EXAMPLE: All facts |Group property non-implications |Subgroup property non-implications |Group metaproperty dissatisfactionsSubgroup metaproperty dissatisfactions

Definition

This group is defined in the following equivalent ways:

1. It is the alternating group of degree eight, i.e., over a set of size eight.
2. It is the projective special linear group of degree four over the field of two elements, i.e., $PSL(4,2)$. It is also the special linear group $SL(4,2)$, the projective general linear group $PGL(4,2)$, and the general linear group $GL(4,2)$.

This is one member of the smallest order pair of non-isomorphic finite simple non-abelian groups having the same order. The other member of this pair is projective special linear group:PSL(3,4).

Arithmetic functions

Function	Value	Similar groups	Explanation
order (number of elements, equivalently, cardinality or size of underlying set)	20160	groups with same order	As alternating group: $8!/2=(8\cdot 7\cdot 6\cdot 5\cdot 4\cdot 2\cdot 3\cdot 1)/2=20160$ As general linear group: $(2^4-1)(2^4-2)(2^4-2^2)(2^4-2^3)=15\cdot 14\cdot 12\cdot 8=20160$

GAP implementation

Description	Functions used
AlternatingGroup(8)	AlternatingGroup
PSL(4,2)	PSL
SL(4,2)	SL
PGL(4,2)	PGL
GL(4,2)	GL