Alternating group:A5

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Contents

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:

1. It is the group of even permutations (viz., the alternating group) on five elements.
2. It is the von Dyck group (sometimes termed triangle group, though the latter has a slightly different meaning) with parameters $(2,3,5)$ (sometimes written in reverse order as $(5,3,2)$).
3. 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
4. It is the projective special linear group of degree two over the field of four elements, viz., $PSL(2,4)$. It is also the special linear group of degree two over the field of four elements, i.e., $SL(2,4)$. It is also the projective general linear group of degree two over the field of four elements, i.e., $PGL(2,4)$.
5. It is the projective special linear group of degree two over the field of five elements, viz., $PSL(2,5)$.

Equivalence of definitions

• The equivalence of the definitions within (4) is given by isomorphism between linear groups when degree power map is bijective.
• PGL(2,4) is isomorphic to A5: equivalence of (1) and (4)
• PSL(2,5) is isomorphic to A5: equivalence of (1) and (5)

IMPORTANT NOTE: This page concentrates on $A_5$ as an abstract group in its own right. To learn more about this group as a subgroup of index two inside symmetric group:S5, see A5 in S5.

Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 60#Arithmetic functions

Basic arithmetic functions

Function	Value	Similar groups	Explanation
order (number of elements, equivalently, cardinality or size of underlying set)	60	groups with same order	As $A_n, n = 5$, $n!/2 = 5!/2 = 60$ As $SL(2,q), q = 4$, $q^3 - q = 4^3 - 4 = 60$ As $PSL(2,q), q = 5$, $(q^3 - q)/\operatorname{gcd}(2,q-1) = (5^3 - 5)/2 = 60$. As von Dyck group with parameters $(p,q,r) = (2,3,5)$: $\frac{2}{1/p + 1/q + 1/r - 1} = \frac{2}{1/2 + 1/3 + 1/5 - 1} = 60$ See element structure of alternating group:A5#Order computation for more.
exponent of a group	30	groups with same order and exponent of a group | groups with same exponent of a group	As $A_n, n = 5$, $n$ odd: $\operatorname{lcm} \{ 1,2,\dots,n - 2,n \}$ (exclude $n - 1$): becomes $\operatorname{lcm}\{1,2,3,5 \} = 30$ As $SL(2,q), q = 4$ (power of 2): $2(q^2 - 1) = 2(4^2 - 1) = 30$ As $PSL(2,q), q = 5, p = 5$ (odd): $p(q^2 - 1)/4 = 5(5^2 - 1)/4 = 5(6) = 30$
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 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	$(1,2,3), (1,2,3,4,5)$.
subgroup rank of a group	2	groups with same order and subgroup rank of a group | groups with same subgroup rank of a group	--
max-length of a group	4	groups with same order and max-length of a group | groups with same max-length of a group	--
composition length	1	groups with same order and composition length | groups with same composition length	--
chief length	1	groups with same order and chief length | groups with same chief length	--

Arithmetic functions of a counting nature

Function	Value	Explanation
number of subgroups	59	subgroup structure of alternating group:A5
number of conjugacy classes	5	As $A_n, n = 5$: (2 * (number of self-conjugate partitions of 5)) + (number of conjugate pairs of non-self-conjugate partitions of 5) = $2(1) + 3 = 5$ (more here) As $PSL(2,q)$, $q = 5$: $(q + 5)/2 = (5 + 5)/2 = 5$ (more here) As $PGL(2,q)$, $q = 4$ (even): $q + 1 = 4 + 1 = 5$ (more here) See element structure of alternating group:A5#Number of conjugacy classes for more
number of conjugacy classes of subgroups	9	subgroup structure of alternating group:A5

Group properties

Basic properties

Property	Satisfied	Explanation	Comment
abelian group	No	$(1,2,3)$, $(1,2,3,4,5)$ don't commute	$A_n$ is non-abelian, $n \ge 4$.
nilpotent group	No	Centerless: The center is trivial	$A_n$ is non-nilpotent, $n \ge 4$.
metacyclic group	No	Simple and non-abelian	$A_n$ is not metacyclic, $n \ge 4$.
supersolvable group	No	Simple and non-abelian	$A_n$ is not supersolvable, $n \ge 4$.
solvable group	No		$A_n$ is not solvable, $n \ge 5$.
simple group	Yes	Smallest simple non-abelian group

Other properties

Property	Satisfied	Explanation	Comment
T-group	Yes	Simple and non-abelian
rational-representation group	No
rational group	No
ambivalent group	Yes		Also see classification of ambivalent alternating groups
complete group	No	Conjugation by odd permutations of $S_5$ gives outer automorphisms
perfect group	Yes	Follows from it being a simple non-abelian group.
Schur-trivial group	No	The Schur multiplier is cyclic group:Z2.
N-group	Yes	See classification of alternating groups that are N-groups	$A_n$ is a N-group only for $n \le 7$.

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

Further information: element structure of alternating group:A5

Summary

Item	Value
order of the whole group (total number of elements)	60 prime factorization $2^2 \cdot 3^1 \cdot 5^1= 4 \cdot 3 \cdot 5$ See order computation for more
conjugacy class sizes	1,12,12,15,20 maximum: 20, number: 5, sum (equals order of group): 60, lcm: 60 See conjugacy class structure for more.
number of conjugacy classes	5 See element structure of alternating group:A5#Number of conjugacy classes
order statistics	1 of order 1, 15 of order 2, 20 of order 3, 24 of order 5 maximum: 5, lcm (exponent of the whole group): 30

Conjugacy class structure

For a symmetric group, cycle type determines conjugacy class. The statement is almost true for the alternating group, except for the fact that some conjugacy classes of even permutations in the symmetric group split into two in the alternating group, as per the splitting criterion for conjugacy classes in the alternating group, which says that a conjugacy class of even permutations splits in the alternating group if and only if it is the product of odd cycles of distinct length.

Here are the unsplit conjugacy classes:

Partition	Verbal description of cycle type	Representative element of the cycle type	All elements of the cycle type	Size of conjugacy class	Formula for size	Element order
1 + 1 + 1 + 1 + 1	five fixed points	$()$ -- the identity element	$()$	1	$\! \frac{5!}{(1)^5(5!)}$	1
3 + 1 + 1	one 3-cycle, two fixed points	$(1,2,3)$	[SHOW MORE] $(1,2,3)$, $(1,3,2)$, $(1,2,4)$, $(1,4,2)$, $(1,2,5)$, $(1,5,2)$, $(1,3,4)$, $(1,4,3)$, $(1,3,5)$, $(1,5,3)$, $(1,4,5)$, $(1,5,4)$, $(2,3,4)$, $(2,4,3)$, $(2,3,5)$, $(2,5,3)$, $(2,4,5)$, $(2,5,4)$, $(3,4,5)$, $(3,5,4)$	20	$\! \frac{5!}{(3)(1)^2(2!)}$	3
2 + 2 + 1	double transposition: two 2-cycles, one fixed point	$(1,2)(3,4)$	[SHOW MORE] $(1,2)(3,4)$, $(1,3)(2,4)$, $(1,4)(2,3)$, $(2,3)(4,5)$, $(2,4)(3,5)$, $(2,5)(3,4)$, $(1,3)(4,5)$, $(1,4)(3,5)$, $(1,5)(3,4)$, $(1,2)(4,5)$, $(1,4)(2,5)$, $(1,5)(2,4)$, $(1,2)(3,5)$, $(1,3)(2,5)$, $(1,5)(2,3)$	15	$\! \frac{5!}{(2)^2(2!)(1)}$	2

Here is the split pair of conjugacy classes:

Partition	Verbal description of cycle type	Combined size of conjugacy classes	Formula for combined size	Size of each half	Representative of first half	Representative of second half	Real?	Rational?	Element order
5	one 5-cycle	24	$\! \frac{5!}{5}$	12	$(1,2,3,4,5)$	$(1,3,5,2,4)$	Yes	No	5

Up to 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

Further information: Subgroup structure of alternating group:A5

Quick summary

Item	Value
number of subgroups	59 Compared with $A_n, n = 3,4,5,\dots$: 2, 10, 59, 501, 3786, 48337, ...
number of conjugacy classes of subgroups	9 Compared with $A_n, n = 3,4,5,\dots$: 2, 5, 9, 22, 40, 137, ...
number of automorphism classes of subgroups	9 Compared with $A_n, n = 3,4,5,\dots$: 2, 5, 9, 16, 37, 112, ...
isomorphism classes of Sylow subgroups and the corresponding fusion systems	2-Sylow: Klein four-group (order 4) as V4 in A5 (with its simple fusion system -- see simple fusion system for Klein four-group). Sylow number is 5. 3-Sylow: cyclic group:Z3 (order 3) as Z3 in A5. Sylow number is 10. 5-Sylow: cyclic group:Z5 (order 5) as Z5 in A5. Sylow number is 6.
Hall subgroups	In addition to the whole group, trivial subgroup, and Sylow subgroups: $\{ 2,3 \}$-Hall subgroup of order 12 (A4 in A5). There is no $\{ 2,5 \}$-Hall subgroup or $\{3,5\}$-Hall subgroup.
maximal subgroups	maximal subgroups have orders 6 (twisted S3 in A5), 10 (D10 in A5), 12 (A4 in A5)
normal subgroups	only the whole group and the trivial subgroup, because the group is simple. See alternating groups are simple.

Table classifying subgroups up to automorphisms

Note that A5 is simple, and hence no proper nontrivial subgroup is normal or subnormal.

Automorphism class of subgroups	Representative subgroup (full list if small, generating set if large)	Isomorphism class	Order of subgroups	Index of subgroups	Number of conjugacy classes	Size of each conjugacy class	Total number of subgroups	Note
trivial subgroup	$()$	trivial group	1	60	1	1	1	trivial
subgroup generated by double transposition in A5	$\{ (), (1,2)(3,4) \}$	cyclic group:Z2	2	30	1	15	15
V4 in A5	$\{ (), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3) \}$	Klein four-group	4	15	1	5	5	2-Sylow
A3 in A5	$\{ (), (1,2,3), (1,3,2) \}$	cyclic group:Z3	3	20	1	10	10	3-Sylow
twisted S3 in A5	$\langle (1,2,3), (1,2) (4,5)\rangle$	symmetric group:S3	6	10	1	10	10	maximal
A4 in A5	$\langle (1,2)(3,4), (1,2,3) \rangle$	alternating group:A4	12	5	1	5	5	2,3-Hall, maximal
Z5 in A5	$\langle (1,2,3,4,5) \rangle$	cyclic group:Z5	5	12	1	6	6	5-Sylow
D10 in A5	$\langle (1,2,3,4,5), (2,5)(3,4) \rangle$	dihedral group:D10	10	6	1	6	6	maximal
whole group	$\langle (1,2,3,4,5), (1,2,3) \rangle$	alternating group:A5	60	1	1	1	1
Total	--	--	--	--	9	--	59	--

Linear representation theory

Further information: linear representation theory of alternating group:A5

Summary

Item	Value
Degrees of irreducible representations over a splitting field (such as $\overline{\mathbb{Q}}$ or $\mathbb{C}$)	1,3,3,4,5 grouped form: 1 (1 time), 3 (2 times), 4 (1 time), 5 (1 time) maximum: 5, lcm: 60, number: 5, sum of squares: 60, quasirandom degree: 3
Schur index values of irreducible representations	1,1,1,1,1
Ring generated by character values	$\mathbb{Z}[(1 + \sqrt{5})/2]$ or $\mathbb{Z}[2\cos(2\pi/5)]$
Minimal splitting field, i.e., smallest field of realization for all irreducible representations	$\mathbb{Q}(\sqrt{5})$ -- quadratic extension of field of rational numbers Same as field generated by character values
Orbit structure under action of automorphism group	orbits of size 1 each of degree 1, 4, and 5 representations, orbit of size 2 of degree 3 representations (interchanged via an automorphism induced by conjugation by odd permutation)
Orbit structure under action of Galois group over rationals	orbits of size 1 each of degree 1, 4, and 5 representations, orbit of size 2 of degree 3 representations (interchanged via the mapping $\sqrt{5} \mapsto -\sqrt{5}$)
Degrees of irreducible representations over the field of rational numbers	1,4,5,6

Character table

Representation/conjugacy class representative and size	$()$ (size 1)	$(1,2)(3,4)$ (size 15)	$(1,2,3)$ (size 20)	$(1,2,3,4,5)$ (size 12)	$(1,2,3,5,4)$ (size 12)
trivial	1	1	1	1	1
restriction of standard	4	0	1	-1	-1
irreducible five-dimensional	5	1	-1	0	0
one irreducible constituent of restriction of exterior square of standard	3	-1	0	$(\sqrt{5} +1)/2$	$(-\sqrt{5} + 1)/2$
other irreducible constituent of restriction of exterior square of standard	3	-1	0	$(-\sqrt{5} + 1)/2$	$(\sqrt{5} + 1)/2$

Supergroups

Further information: supergroups of alternating group:A5

Subgroups: making all automorphisms inner

Further information: symmetric group:S5, A5 in S5

The outer automorphism group of alternating group:A5 is cyclic group:Z2, and the whole automorphism group is symmetric group:S5. Since alternating group:A5 is a centerless group, it embeds as a subgroup of index two inside its automorphism group, which is symmetric group on five elements.

$A_5$ is a simple non-abelian group and $A_5$ and $S_5$ are the only two almost simple groups corresponding to $A_5$.

$A_5$ is also of index two in the full icosahedral group, which turns out not to be $S_5$, but instead the direct product of $A_5$ and the cyclic group of order two.

Quotients: Schur covering groups

Further information: Group cohomology of alternating group:A5#Schur multiplier, group cohomology of alternating groups, double cover of alternating group

Further information: special linear group:SL(2,5), center of special linear group:SL(2,5)

The Schur multiplier of $A_5$ is cyclic group:Z2.

The corresponding universal central extension (the unique Schur covering group, unique because $A_5$ is a perfect group) is special linear group:SL(2,5), also denoted as $2 \cdot A_5$ to denote that it is a double cover (see double cover of alternating group). The center of special linear group:SL(2,5) is cyclic group:Z2 and the quotient group is $A_5$.

$A_5$ is a simple non-abelian group and $A_5$ and $SL(2,5) = 2 \cdot A_5$ are the only two corresponding quasisimple groups.

External links

• A5 on the Atlas of Finite Groups

GAP implementation

Group ID

This finite group has order 60 and has ID 5 among the groups of order 60 in GAP's SmallGroup library. For context, there are 13 groups of order 60. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(60,5)

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

gap> G := SmallGroup(60,5);

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

IdGroup(G) = [60,5]

or just do:

IdGroup(G)

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

Equivalent command: Running this command constructs the group as AlternatingGroup(5).

Memory usage: The memory usage for the SmallGroup construction is 1452.

Other descriptions

Description	Functions used	Output storage format (verification command)	Memory usage (can be computed by invoking the MemoryUsage function)	Groups containing this as a subgroup
AlternatingGroup(5)	AlternatingGroup	permutation group (IsPermGroup)	194	AlternatingGroup or SymmetricGroup of degree at least five.
PSL(2,4)	PSL	permutation group (IsPermGroup)	2289	itself
SL(2,4)	SL	matrix group (IsMatrixGroup)	1194	GL(2,q), for $q$ a power of 4.
PGL(2,4)	PGL	permutation group (IsPermGroup)	2289	itself
PSL(2,5)	PSL	permutation group (IsPermGroup)	2125	PGL(2,5)
PerfectGroup(60) or equivalently PerfectGroup(60,1)	PerfectGroup	finitely presented group (IsFpGroup)	233	--
SimpleGroup("Alt",5)	SimpleGroup	permutation group (IsPermGroup)	229	--
SmallSimpleGroup(60)	SmallSimpleGroup	permutation group (IsPermGroup)	229	--
AllSmallNonabelianSimpleGroups([1..100])[1]	AllSmallNonabelianSimpleGroups	permutation group (IsPermGroup)	229	--