Alternating group:A6

This article is about a particular group, i.e., a group unique upto isomorphism. View specific information (such as linear representation theory, subgroup structure) about this group
View a complete list of particular groups (this is a very huge list!)[SHOW MORE]

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

This particular group is a finite group of order: 360 This particular group is the smallest (in terms of order): simple non-Abelian group that is not a minimal simple group This particular group is the smallest (in terms of order): simple non-Abelian group that is neither complete nor maximal in its automorphism group

Definition

The alternating group $A_{6}$ is defined in the following equivalent ways:

• It is the group of even permutations (viz., the alternating group) on six elements.
• It is the projective special linear group $PSL(2,9)$.

Arithmetic functions

Function	Value	Explanation
order	360	$6!/2=360$.
exponent	60	Elements of order $3,4,5$.
derived length	--	not a solvable group.
nilpotency class	--	not a nilpotent group.
Frattini length	1	Frattini-free group: intersection of maximal subgroups is trivial.
minimum size of generating set	2
subgroup rank	2	--
max-length	5	--

Group properties

Property	Satisfied	Explanation	Comment
Abelian group	No	$(1,2,3)$, $(1,2,3,4,5)$ don't commute	$A_{n}$ is non-abelian, $n\geq 4$.
Nilpotent group	No	Centerless: The center is trivial	$A_{n}$ is non-nilpotent, $n\geq 4$.
Metacyclic group	No	Simple and non-abelian	$A_{n}$ is not metacyclic, $n\geq 4$.
Supersolvable group	No	Simple and non-abelian	$A_{n}$ is not supersolvable, $n\geq 4$.
Solvable group	No		$A_{n}$ is not solvable, $n\geq 5$.
Simple non-abelian group	Yes	alternating groups are simple, projective special linear group is simple
T-group	Yes	Simple and non-abelian