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Math and science::Algebra::Aluffi

Cosets

Coset

Let \( G \) be a group and \( H \subset G \) a subgroup.

For any \( a \in G \), a right-coset of \( H \) is any set of the form:

[\[ ? \]]

For any \( a \in G, \) a left-coset of \( H \) is any set of the form:

[\[ ? \]]

Coset equivalence relation

A right-coset forms a block in a partition defined by an equivalence relation, \( \sim_{R} \).

[\[ x \sim_{R} y \quad \iff \quad ? \]]

A left-coset forms a block in a partition defined by a equivalence relation, \( \sim_{L} \).

[\[ x \sim_{L} y \quad \iff \quad ? \]]

The essence of the relations is to consider an element related to another if the inverse of one element brings the other element back to \( H \).