Alternating group:A5
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Contents
Definition
The alternating group , also denoted , 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 (sometimes written in reverse order as ).
- 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 or . 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 degree two over the field of four elements, viz., . It is also the special linear group of degree two over the field of four elements, i.e., . It is also the projective general linear group of degree two over the field of four elements, i.e., .
- It is the projective special linear group of degree two over the field of five elements, viz., .
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 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
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 : (2 * (number of self-conjugate partitions of 5)) + (number of conjugate pairs of non-self-conjugate partitions of 5) = (more here) As , : (more here) As , (even): (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 | , don't commute | is non-abelian, . |
nilpotent group | No | Centerless: The center is trivial | is non-nilpotent, . |
metacyclic group | No | Simple and non-abelian | is not metacyclic, . |
supersolvable group | No | Simple and non-abelian | is not supersolvable, . |
solvable group | No | is not solvable, . | |
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 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 | is a N-group only for . |
Endomorphisms
Automorphisms
The automorphism group of is the symmetric group on five letters , with embedded in it as inner automorphisms.
Concretely, we can think of as embedded in , and acting on by conjugation. The automorphisms obtained this way are all the automorphisms of .
Other endomorphisms
Since is a finite simple group, it is a group in which every endomorphism is trivial or an automorphism. In particular, the endomorphisms of 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 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 | 1 | ||
3 + 1 + 1 | one 3-cycle, two fixed points | [SHOW MORE] | 20 | 3 | ||
2 + 2 + 1 | double transposition: two 2-cycles, one fixed point | [SHOW MORE] | 15 | 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 | 12 | 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 .
Subgroups
Further information: Subgroup structure of alternating group:A5
Quick summary
Item | Value |
---|---|
number of subgroups | 59 Compared with : 2, 10, 59, 501, 3786, 48337, ... |
number of conjugacy classes of subgroups | 9 Compared with : 2, 5, 9, 22, 40, 137, ... |
number of automorphism classes of subgroups | 9 Compared with : 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: -Hall subgroup of order 12 (A4 in A5). There is no -Hall subgroup or -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 | cyclic group:Z2 | 2 | 30 | 1 | 15 | 15 | ||
V4 in A5 | Klein four-group | 4 | 15 | 1 | 5 | 5 | 2-Sylow | |
A3 in A5 | cyclic group:Z3 | 3 | 20 | 1 | 10 | 10 | 3-Sylow | |
twisted S3 in A5 | symmetric group:S3 | 6 | 10 | 1 | 10 | 10 | maximal | |
A4 in A5 | alternating group:A4 | 12 | 5 | 1 | 5 | 5 | 2,3-Hall, maximal | |
Z5 in A5 | cyclic group:Z5 | 5 | 12 | 1 | 6 | 6 | 5-Sylow | |
D10 in A5 | dihedral group:D10 | 10 | 6 | 1 | 6 | 6 | maximal | |
whole group | 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 or ) | 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 | or |
Minimal splitting field, i.e., smallest field of realization for all irreducible representations | -- 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 ) |
Degrees of irreducible representations over the field of rational numbers | 1,4,5,6 |
Character table
Representation/conjugacy class representative and size | (size 1) | (size 15) | (size 20) | (size 12) | (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 | ||
other irreducible constituent of restriction of exterior square of standard | 3 | -1 | 0 |
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.
is a simple non-abelian group and and are the only two almost simple groups corresponding to .
is also of index two in the full icosahedral group, which turns out not to be , but instead the direct product of 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 is cyclic group:Z2.
The corresponding universal central extension (the unique Schur covering group, unique because is a perfect group) is special linear group:SL(2,5), also denoted as 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 .
is a simple non-abelian group and and are the only two corresponding quasisimple groups.
External links
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 :
gap> G := SmallGroup(60,5);
Conversely, to check whether a given group 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 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 | -- |