Comparison of cosets and conjugacy classes ↩

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created: 2021-12-17 17:57:40
modified: 2022-01-10 04:13:03

Cosets	Conjugacy classes
To define them we need a subgroup in the group . Left and right cosets are not the same unless the subgroup is normal.	The group itself defines them.
The cardinality of each coset is the same, equal to the order of . Also the order of divides the order of due to Lagrange's theorem. Therefore the number of cosets also divides the order of .	Each conjugacy class may be of different size (e.g. is always alone in its class, ). The size of each conjugacy class also divides the order of .
They can be thought of as equivalence classes.	They can be thought of as equivalence classes.
They cover the whole group.	They cover the whole group.
If the subgroup is normal, the quotient group is automatically defined, together with the natural homomorphism.