Class Equation #
This file establishes the class equation for finite groups.
Main statements #
Group.card_center_add_sum_card_noncenter_eq_card
: The class equation for finite groups. The cardinality of a group is equal to the size of its center plus the sum of the size of all its nontrivial conjugacy classes. AlsoGroup.nat_card_center_add_sum_card_noncenter_eq_card
.
theorem
sum_conjClasses_card_eq_card
(G : Type u_1)
[Group G]
[Fintype (ConjClasses G)]
[Fintype G]
[(x : ConjClasses G) → Fintype ↑x.carrier]
:
(Finset.univ.sum fun (x : ConjClasses G) => x.carrier.toFinset.card) = Fintype.card G
Conjugacy classes form a partition of G, stated in terms of cardinality.
theorem
Group.sum_card_conj_classes_eq_card
(G : Type u_1)
[Group G]
[Finite G]
:
(finsum fun (x : ConjClasses G) => x.carrier.ncard) = Nat.card G
Conjugacy classes form a partition of G, stated in terms of cardinality.
theorem
Group.nat_card_center_add_sum_card_noncenter_eq_card
(G : Type u_1)
[Group G]
[Finite G]
:
(Nat.card ↥(Subgroup.center G) + finsum fun (x : ConjClasses G) => finsum fun (h : x ∈ ConjClasses.noncenter G) => Nat.card ↑x.carrier) = Nat.card G
The class equation for finite groups. The cardinality of a group is equal to the size of its center plus the sum of the size of all its nontrivial conjugacy classes.
theorem
Group.card_center_add_sum_card_noncenter_eq_card
(G : Type u_2)
[Group G]
[(x : ConjClasses G) → Fintype ↑x.carrier]
[Fintype G]
[Fintype ↥(Subgroup.center G)]
[Fintype ↑(ConjClasses.noncenter G)]
:
(Fintype.card ↥(Subgroup.center G) + (ConjClasses.noncenter G).toFinset.sum fun (x : ConjClasses G) => x.carrier.toFinset.card) = Fintype.card G