Alternating group:A7
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Contents
Definition
This group is defined as the alternating group of degree , i.e., the alternating group on a set of size . In other words, it is the subgroup of symmetric group:S7 comprising the even permutations.
Arithmetic functions
Basic arithmetic functions
Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 2520#Arithmetic functions
Function | Value | Similar groups | Explanation |
---|---|---|---|
order (number of elements, equivalently, cardinality or size of underlying set) | 2520 | groups with same order | As alternating group : which simplifies to |
exponent of a group | 420 | groups with same order and exponent of a group | groups with same exponent of a group | |
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 all 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 |
Arithmetic functions of a counting nature
Group 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 non-abelian group | Yes | alternating groups are simple, projective special linear group is simple | |
T-group | Yes | Simple and non-abelian | |
Ambivalent group | No | Classification of ambivalent alternating groups | |
Rational-representation group | No | ||
Rational group | No | ||
Complete group | No | Conjugation by odd permutations in gives outer automorphisms. | |
N-group | Yes | See classification of alternating groups that are N-groups | is a N-group only for . |
Elements
Further information: element structure of alternating group:A7
Subgroups
Further information: subgroup structure of alternating group:A7
Quick summary
Item | Value |
---|---|
Number of subgroups | 3786 Compared with : 2, 10, 59, 501, 3786, 48337, ... |
Number of conjugacy classes of subgroups | 40 Compared with : 2, 5, 9, 22, 40, 137, ... |
Number of automorphism classes of subgroups | 37 Compared with : 2, 5, 9, 16, 37, 112, ... |
Isomorphism classes of Sylow subgroups and the corresponding fusion systems | 2-Sylow: dihedral group:D8 (order 8) as D8 in A7 (with its non-inner fusion system -- see fusion systems for dihedral group:D8). Sylow number is 315. 3-Sylow: elementary abelian group:E9 (order 9) as E9 in A7. Sylow number is 70. 5-Sylow: cyclic group:Z5 (order 5) as Z5 in A7. Sylow number is 126. 7-Sylow: cyclic group:Z7 (order 7) as Z7 in A7. Sylow number is 120. |
Hall subgroups | Other than the whole group, the trivial subgroup, and the Sylow subgroups, there are -Hall subgroups (of order 72) and -Hall subgroups (of order 360), the latter being A6 in A7. Note that the -Hall subgroups are not contained in -Hall subgroups. |
maximal subgroups | maximal subgroups have orders 72, 120, 168, 360. |
normal subgroups | only the whole group and the trivial subgroup, because the group is simple. See alternating groups are simple. |
subgroups that are simple non-abelian groups (apart from the whole group itself) | alternating group:A5 (order 60), projective special linear group:PSL(3,2) (order 168, the embedding arises via the natural permutation representation on the two-dimensional projective space over field:F2, which has size ), alternating group:A6 (order 360) |
Linear representation theory
Further information: linear representation theory of alternating group:A7
Item | Value |
---|---|
degrees of irreducible representations over a splitting field | 1,6,10,10,14,14,15,21,35 grouped form: 1 (1 time), 6 (1 time), 10 (2 times), 14 (2 times), 15 (1 time), 21 (1 time), 35 (1 time) maximum: 35, lcm: 210, number: 9, sum of squares: 2520 |
Minimal splitting field, i.e., field of realization of all irreducible representations (characteristic zero) | where is a primitive seventh root of unity. This is the same as Quadratic extension of Same as field generated by character values |
Condition for a field of characteristic not 2,3,5, or 7, to be a splitting field | -7 should be a square in the field. |
Minimal splitting field, i.e., field of realization of all irreducible representations in prime characteristic | Case : prime field Case : quadratic extension |
Smallest size splitting field | field:F11 |
GAP implementation
Description | Functions used |
---|---|
AlternatingGroup(7) | AlternatingGroup |
PerfectGroup(2520) or PerfectGroup(2520,1) | PerfectGroup |
SimpleGroup("Alt",7) | SimpleGroup |