Alternating group:A5

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 the smallest (in terms of order): simple non-Abelian group

This particular group is the smallest (in terms of order): non-solvable group

This particular group is the smallest (in terms of order): nontrivial perfect group

This particular group is a finite group of order: 60

Definition

The alternating group $A_{5}$, also denoted $\operatorname {Alt} (5)$, and termed the alternating group of degree five, is defined in the following ways:

• It is the group of even permutations (viz, the alternating group) on five elements.
• It is the von Dyck group (sometimes termed triangle group, though the latter has a slightly different meaning) with parameters $(5,3,2)$.
• It is the icosahedral group, i.e., the group of orientation-preserving symmetries of the regular icosahedron (or equivalently, regular dodecahedron). Viewed this way, it is denoted $l$ or $532$. Further information: Classification of finite subgroups of SO(3,R), Linear representation theory of alternating group:A5
• It is the projective special linear group of order two over the field of four elements, viz., $PSL(2,4)$.
• It is the projective special linear group of order two over the field of five elements, viz., $PSL(2,5)$.

Arithmetic functions

Function	Value	Explanation
order	60	$5!/2=60$.
exponent	30	Elements of order $2,3,5$.
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	$(1,2,3),(1,2,3,4,5)$.
subgroup rank	2	--
max-length	4	--

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	Smallest simple non-abelian group
T-group	Yes	Simple and non-abelian

Endomorphisms

Automorphisms

The automorphism group of $A_{5}$ is the symmetric group on five letters $S_{5}$, with $A_{5}$ embedded in it as inner automorphisms.

Concretely, we can think of $A_{5}$ as embedded in $S_{5}$, and $S_{5}$ acting on $A_{5}$ by conjugation. The automorphisms obtained this way are all the automorphisms of $A_{5}$.

Other endomorphisms

Since $A_{5}$ is a finite simple group, it is a group in which every endomorphism is trivial or an automorphism. In particular, the endomorphisms of $A_{5}$ are: the trivial map, and the automorphisms described above.

Elements

Upto conjugacy

Further information: Splitting criterion for conjugacy classes in the alternating group

There are the following conjugacy classes. These correspond to the ways of partitioning $5$ as sums of numbers where the number of even numbers is even; the partitions where all the parts are odd and distinct give rise to two conjugacy classes:

1. $5=1+1+1+1+1$, five fixed points: The identity element. (1)
2. $5=2+2+1$, two transpositions and one fixed point: The conjugacy class of double transpositions, such as $(1,2)(3,4)$. (15)
3. $5=3+1+1$, one 3-cycle and two fixed points: The conjugacy class of 3-cycles. (20)
4. $5=5$: The conjugacy class of 5-cycles conjugate to $(1,2,3,4,5)$. (12)
5. $5=5$: The conjugacy class of 5-cycles conjugate to $(1,3,5,2,4)$. (12)

Upto automorphism

Under outer automorphisms, the fourth and fifth conjugacy classes get merged. Thus, the classes under automorphism are of size $1,15,20,24$.

Subgroups

Here is a complete list of subgroups of $A_{5}$.

1. The trivial subgroup. (1)
2. The two-element subgroup generated by a double transposition, such as $(1,2)(3,4)$. Isomorphic to Cyclic group:Z2.(15)
3. The four-element subgroup comprising all double transpositions on four of the five elements, such as $\langle (),(1,2)(3,4),(1,3)(2,4),(1,4)(2,3)\rangle$. Isomorphic to Klein four-group.(5)
4. The three-element subgroup generated by a 3-cycle, such as $(1,2,3)$. Isomorphic to cyclic group:Z3.(10)
5. A six-element subgroup that is isomorphic to the symmetric group on three letters. This moves three elements as the symmetric group on those three elements does, while it transposes the other two elements iff it is odd. For instance, $\langle (1,2,3),(1,3,2),(),(1,2)(4,5),(2,3)(4,5),(1,3)(4,5)\rangle$. (10)
6. A twelve-element subgroup that is the alternating group on four of the five letters. (5)
7. A five-element subgroup generated by a 5-cycle. Isomorphic to cyclic group:Z5. (6)
8. A ten-element subgroup generated by a 5-cycle and a double transposition that conjugates it to its inverse. Isomorphic to dihedral group:D10. (6)
9. The whole group. (1)

Normal subgroups

The normal subgroups of $A_{5}$ are the whole group and the trivial subgroup.

Characteristic subgroups

In this group, the characteristic subgroups are the same as the normal subgroups. In other words, this is a group in which every normal subgroup is characteristic

Subnormal subgroups

The subnormal subgroups are the same as the normal subgroups: the whole group and the trivial subgroup.

Permutable subgroups

The permutable subgroups are the same as the normal subgroups: the whole group and the trivial subgroup.

Abnormal subgroups

The groups of types (7), (8), (10) and (11) are all abnormal.

Self-normalizing subgroups

The self-normalizing subgroups are the same as the abnormal subgroups.

Bigger groups

Groups having it as a subgroup

The alternating group is a subgroup of index two inside the symmetric group on five elements. It is also of index two in the full icosahedral symmetry group, which turns out not to be $S_{5}$, but instead the direct product of $A_{5}$ and the cyclic group of order two.

Groups having it as a quotient

The alternating group is a quotient of $SL(2,5)$ by its center. Hence, it is the inner automorphism group of $SL(2,5)$. $SL(2,5)$ is also the universal central extension of the alternating group.