Alternating group:A7

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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

Definition

This group is defined as the alternating group of degree $7$ , i.e., the alternating group on a set of size $7$ . 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 $A_n, n = 7$ : $n!/2 = 7!/2$ which simplifies to $(7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1)/2 = 2520$
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

Function	Value	Explanation
number of subgroups	3786	See subgroup structure of alternating group:A7, subgroup structure of alternating groups
number of conjugacy classes	9	See element structure of alternating group:A7, element structure of alternating groups
number of conjugacy classes of subgroups	40	See subgroup structure of alternating group:A7, subgroup structure of alternating groups

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, $n \ge 4$ .
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 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 $S_6$ gives outer automorphisms.
N-group	Yes	See classification of alternating groups that are N-groups	$A_n$ is a N-group only for $n \le 7$ .

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 $A_n, n = 3,4,5,\dots$ : 2, 10, 59, 501, 3786, 48337, ...
Number of conjugacy classes of subgroups	40 Compared with $A_n, n = 3,4,5,\dots$ : 2, 5, 9, 22, 40, 137, ...
Number of automorphism classes of subgroups	37 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: 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 $\{ 2,3 \}$ -Hall subgroups (of order 72) and $\{ 2,3,5 \}$ -Hall subgroups (of order 360), the latter being A6 in A7. Note that the $\{ 2,3 \}$ -Hall subgroups are not contained in $\{ 2,3,5 \}$ -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 $2^2 + 2 + 1 = 7$ ), 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)	$\mathbb{Q}(\omega + \omega^2 + \omega^4)$ where $\omega$ is a primitive seventh root of unity. This is the same as $\mathbb{Q}(\sqrt{-7})$ Quadratic extension of $\mathbb{Q}$ 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 $p \ne 2,3,5,7$	Case $p \equiv 1,2,4 \pmod 7$ : prime field $\mathbb{F}_p$ Case $p \equiv 3,5,6 \pmod 7$ : quadratic extension $\mathbb{F}_{p^2}$
Smallest size splitting field	field:F11