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).

Equivalence of definitions

The equivalence between the various definitions within (2) follows from isomorphism between linear groups over field:F2.

Arithmetic functions

Basic arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 20160#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 $A_n,n=8$: $n!/2=8!/2=(8\cdot 7\cdot 6\cdot 5\cdot 4\cdot 2\cdot 3\cdot 1)/2=20160$ As general linear group: $(2^4-1)(2^4-2)(2^4-2^2)(2^4-2^3)=15\cdot 14\cdot 12\cdot 8=20160$
exponent of a group	420	groups with same order and exponent of a group | groups with same exponent of a group
derived length	--	--	not a solvable group
nilpotency class	--	--	not a nilpotent group
Frattini length	1	groups with same order and Frattini length | groups with same Frattini length	Frattini-free group: intersection of all maximal subgroups is trivial
minimum size of generating set	2	groups with same order and minimum size of generating set | groups with same minimum size of generating set

Arithmetic functions of a counting nature

Function	Value	Explanation
number of conjugacy classes	14	As $A_n,n=8$: (2 * (number of self-conjugate partitions of 8)) + (number of conjugate pairs of non-self-conjugate partitions of 8) = $(2*2)+10=14$ (more here) As $GL(4,q),q=2$: $q^4-q=2^4-2=14$ (more here) As $SL(4,q),q=2$: $q^3+q^2+q=2^3+2^2+2=14$ (more here) See element structure of alternating group:A8
number of conjugacy classes of subgroups	137	See subgroup structure of alternating group:A8, subgroup structure of alternating groups
number of subgroups	48337	See subgroup structure of alternating group:A8, subgroup structure of alternating groups