Alternating group:A4

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

This particular group is the smallest (in terms of order): group not having subgroups of every order dividing the group order

This particular group is a finite group of order: 12

Definition

The alternating group $A_4$ is defined in the following equivalent ways:

• It is the group of even permutations (viz., the alternating group) on four elements.
• It is the von Dyck group (sometimes termed triangle group) with parameters $(3,3,2)$.
• It is the group of orientation-preserving symmetries of a regular tetrahedron. When viewed in this light, it is called the tetrahedral group, and its symbol as a point group is $T$ or $332$. Further information: Classification of finite subgroups of SO(3,R), Linear representation theory of alternating group:A4
• It is the projective special linear group of degree 2 over the field of three elements, viz., $PSL(2,3)$.
• It is the general affine group of degree $1$ over the field of four elements, viz., $GA(1,4)$ (also written as $AGL(1,4)$.

Arithmetic functions

Function	Value	Explanation
order	12	$4!/2=12$.
exponent	6	Elements of order $2,3$.
derived length	2	Derived series goes through Klein four-group of double transpositions.
nilpotency class	--	not a nilpotent group.
Frattini length	1	Frattini-free group: intersection of maximal subgroups is trivial.
minimum size of generating set	2	$(1,2,3),(1,2)(3,4)$
subgroup rank	2
max-length	3
composition length	3
chief length	2

Group properties

Property	Satisfied	Explanation	Comment
Abelian group	No	$(1,2,3)$, $(1,2)(3,4)$ don't commute	$A_n$ is non-abelian, $n\ge 4$.
Nilpotent group	No	Centerless: The center is trivial	$A_n$ is non-nilpotent, $n\ge 4$.
Metacyclic group	No	No cyclic normal subgroup	$S_n$ is not metacyclic, $n\ge 4$.
Supersolvable group	No	No cyclic normal subgroup	$S_n$ is not supersolvable, $n\ge 4$.
Solvable group	Yes	Length two, commutator subgroup is Klein four-group	Largest $n$ for which $A_n$ is solvable.
T-group	No	Double transposition generates non-normal 2-subnormal subgroup	Only $n$ for which $A_n$ isn't a T-group.
Group having subgroups of all orders dividing the group order	No	No subgroup of order six
Ambivalent group	No		Classification of ambivalent alternating groups
Rational group	No
Rational-representation group	No
Group in which every element is automorphic to its inverse	Yes	Alternating group implies every element is automorphic to its inverse
Group in which any two elements generating the same cyclic subgroup are automorphic	Yes	Alternating group implies any two elements generating the same cyclic subgroup are automorphic
Frobenius group	Yes	Frobenius kernel is Klein four-subgroup, complement is generated by 3-cycle
Camina group	Yes	Commutator subgroup is Klein four-subgroup, other two cosets are conjugacy classes.

Endomorphisms

Automorphisms

The automorphism group of the alternating group of degree four is isomorphic to the symmetric group of degree four. Since the alternating group of degree four is centerless, it embeds as a subgroup inside its automorphism group.

Another way of thinking of this is as follows: in the symmetric group of degree four, the alternating group of degree four is a subgroup of index two, and every automorphism of this subgroup is realized as the restriction to this subgroup of an inner automorphism of the symmetric group.

Endomorphisms

The endomorphisms of the alternating group of degree four are given by the following (i.e., equivalent to one of these up to composition with an automorphism):

• The trivial map.
• The identity map.
• The retraction to a subgroup of order three, with kernel being the Klein four-group comprising the identity and the double transpositions. (All such retractions are equivalent).

Elements

Upto conjugacy

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

The alternating group on $\{1,2,3,4\}$ has four conjugacy classes. Two of these arise from other partitions of $4$ with an even number of cycles of even length, and with either a repetition of length or a cycle of even length. Two of these arise from a partition of $4$ into cycles of distinct odd length.

1. $4=1+1+1+1$, the identity element. (1)
2. $4=2+2$, the three double transpositions: $(1,2)(3,4),(1,3)(2,4),(1,4)(2,3)$. (3)
3. $4=3+1$, four of the $3$-cycles: $(1,2,3),(4,3,2),(3,4,1),(2,1,4)$. (4)
4. $4=3+1$, the remaining four $3$-cycles: $(1,3,2),(4,2,3),(3,1,4),(2,4,1)$. (4)

Upto automorphism

The conjugacy classes (1) and (2) are invariant under all automorphisms.

An outer automorphism interchanges classes (3) and (4). This can be realized, for instance, by viewing the alternating group as a subgroup of the symmetric group of degree four. Any transposition or $4$-cycle in the symmetric group interchanges classes (3) and (4).

Subgroups

Further information: Subgroup structure of alternating group:A4

The alternating group on $\{1,2,3,4\}$ has the following subgroups (clubbed together by conjugacy):

1. The trivial subgroup. (1)
2. Three subgroups of order two, each generated by a double transposition, such as $(1,2)(3,4)$. These are all isomorphic to the cyclic group of order two. (3)
3. A subgroup of order four, comprising the identity element and the three double transpositions: $\{(),(1,2)(3,4),(1,3)(2,4),(1,4)(2,3)\}$. These are all isomorphic to the Klein four-group. (1)
4. Four subgroups of order three, each generated by a $3$-cycle, such as $(1,2,3)$. These are all isomorphic to the cyclic group of order three. (4)
5. The whole group. (1)

There is no subgroup of order $6$. This is the smallest possible order of a group not having subgroups of all orders dividing the group order.

Normal subgroups

Apart from the trivial subgroup and the whole group, there is exactly one normal subgroup, namely the subgroup of order 4 comprising the identity element and the three double transpositions (this is type (3) in the list above).

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

Apart from the whole group and the trivial subgroup, there are four retracts -- the four Sylow 3-subgroups (listed as type (4) above). These all occur as retracts with the kernel being the subgroup formed by the double transpositions.

Supergroups

These are groups containing the alternating group

The alternating group is contained in the symmetric group on 4 elements, as a normal subgroup of index two. It is, in fact, a fully characteristic subgroup. The complement exists as a subgroup, namely that generated by a transposition.

Subgroup-defining functions

Subgroup-defining function	Subgroup type in list	Isomorphism class	Comment
Center	(1)	Trivial group	$A_n$ is a centerless group, $n\ge 4$.
Commutator subgroup	(3)	Klein four-group
Frattini subgroup	(1)	Trivial group
Fitting subgroup	(3)	Klein four-group

Quotient-defining functions

Quotient-defining function	Isomorphism class	Comment
Inner automorphism group	whole group	The group is centerless, so equals its inner automorphism group.
Abelianization	cyclic group:Z3
Fitting quotient	cyclic group:Z3
Frattini quotient	cyclic group:Z3

Extensions

These are groups having the alternating group as a quotient group Perhaps the most important of these is $SL(2,3)$, which is the universal central extension of $PSL(2,3)$. The kernel of the projection mapping is a two-element subgroup, namely the identity matrix and the negative identity matrix.

GAP implementation

Group ID

This finite group has order 12 and has ID 3 among the groups of order 12 in GAP's SmallGroup library. For context, there are groups of order 12. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(12,3)

For instance, we can use the following assignment in GAP to create the group and name it $G$:

gap> G := SmallGroup(12,3);

Conversely, to check whether a given group $G$ is in fact the group we want, we can use GAP's IdGroup function:

IdGroup(G) = [12,3]

or just do:

IdGroup(G)

to have GAP output the group ID, that we can then compare to what we want.

Other definitions

The alternating group can be constructed in many equivalent ways:

• As the alternating group of degree four, using GAP's AlternatingGroup function:

AlternatingGroup(4)

• Using the von Dyck presentation. Here is a sequence of steps:

F := FreeGroup(3);
G := F/[F.1^3, F.2^3, F.3^2, F.1*F.2*F.3]

The output $G$ is the alternating group.

• As the projective special linear group. The command is

  PSL(2,3)