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 $A_{n},n=7$: $n!/2=7!/2==(7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1)/2=2520$
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