Do equivalence classes form a partition?

[Genady]

Equivalence relation R on set X defines equivalence classes in X. Do they form a partition of X?

[KJW]

I believe so, yes. All elements equivalent to a given element are equivalent to each other. Thus, the set X is partitioned into subsets of elements that are equivalent to other elements of the same subset but not equivalent to any elements of the other subsets.

 

Edited December 26, 2024 by KJW

[Genady]

1 hour ago, KJW said:

subsets of elements that are equivalent to other elements of the same subset but not equivalent to any elements of the other subsets.

 

Plus, even without other equivalent elements, each element forms a class of itself.

[KJW]

2 hours ago, Genady said:

Plus, even without other equivalent elements, each element forms a class of itself.

In group theory, the elements h g h^–1 for each and every element h of the group form an equivalence class for element g of the group. If g belongs to the centre of the group, then its equivalence class is g alone. Otherwise, the equivalence class contains multiple elements. Thus, although the equivalence classes of a group partitions the group, the equivalence classes are not necessarily equal in size.

Edited December 26, 2024 by KJW

[Genady]

48 minutes ago, KJW said:

In group theory, the elements h g h^–1 for each and every element h of the group form an equivalence class for element g of the group.

Does it mean, g₁ ~ g when for some h₁, g₁ = h₁ g h₁^-1 ?

Edited December 26, 2024 by Genady

[KJW]

3 minutes ago, Genady said:

Does it mean, g₁ ~ g when for some h₁   g₁ = h₁ g h₁^-1 ?

Yes. And taken over all elements h of the group, the resulting set is an equivalence class called a conjugacy class. For a given h, the mapping g –> h g h^–1 is called an inner automorphism. Subgroups of the group that are invariant to all inner automorphisms of the group are called normal subgroups of the group. Normal subgroups are very important in group theory. Note that all subgroups are normal in abelian groups.

 

[Genady]

24 minutes ago, KJW said:

Yes. And taken over all elements h of the group, the resulting set is an equivalence class called a conjugacy class. For a given h, the mapping g –> h g h^–1 is called an inner automorphism. Subgroups of the group that are invariant to all inner automorphisms of the group are called normal subgroups of the group. Normal subgroups are very important in group theory. Note that all subgroups are normal in abelian groups.

 

I think I'm missing something. If g₁ = h₁ g h₁^-1 and g₂ = h₂ g h₂^-1 then g₁ ~ g and g₂ ~ g, but it is not necessary that g₁ ~ g₂, which they should if this is an equivalence class. 

?

[KJW]

The importance of normal subgroups is the following (which I won't explain at this time):

Let G be a group and N be a normal subgroup of G. Then there exists a quotient group or factor group G/N and a homomorphism that maps G to G/N, and under this homomorphism, N maps to the identity of G/N. Conversely, for every homomorphism of group G, there exists a normal subgroup N such that N maps to the identity of the image of the homomorphism, and a quotient group or factor group G/N that is isomorphic to the image of the homomorphism.

 

27 minutes ago, Genady said:

I think I'm missing something. If g₁ = h₁ g h₁^-1 and g₂ = h₂ g h₂^-1 then g₁ ~ g and g₂ ~ g, but it is not necessary that g₁ ~ g₂, which they should if this is an equivalence class. 

?

g₁ = h₁ g h₁^–1
g₂ = h₂ g h₂^–1

h₂^–1 g₂ h₂ = g

g₁ = h₁ g h₁^–1 = h₁ h₂^–1 g₂ h₂ h₁^–1 = (h₁ h₂^–1) g₂ (h₁ h₂^–1)^–1

 

Note that in general, (ab)^–1 = b^–1a^–1

 

Edited December 26, 2024 by KJW