# Group coset

Given a subgroup $$H$$ of group $$G$$, the left cosets of $$H$$ in $$G$$ are sets of the form $$\{ gh : h \in H \}$$, for some $$g \in G$$. This is written $$gH$$ as a shorthand.

Similarly, the right cosets are the sets of the form $$Hg = \{ hg: h \in H \}$$.

# Examples

knows-requisite(Symmetric group):

## Symmetric group

In $$S_3$$, the symmetric group on three elements, we can list the elements as $$\{ e, (123), (132), (12), (13), (23) \}$$, using cycle notation. Define $$A_3$$ (which happens to have a name: the alternating group) to be the subgroup with elements $$\{ e, (123), (132) \}$$.

Then the coset $$(12) A_3$$ has elements $$\{ (12), (12)(123), (12)(132) \}$$, which is simplified to $$\{ (12), (23), (13) \}$$.

The coset $$(123)A_3$$ is simply $$A_3$$, because $$A_3$$ is a subgroup so is closed under the group operation.$$(123)$$ is already in $$A_3$$. <div>

more examples, with different requirements

# Properties

• The left cosets of $$H$$ in $$G$$ partition $$G$$. (Proof.)

• For any pair of left cosets of $$H$$, there is a bijection between them; that is, all the cosets are all the same size. (Proof.)

# Why are we interested in cosets?

Under certain conditions (namely that the subgroup $$H$$ must be normal), we may define the quotient group, a very important concept; see the page on “left cosets partition the parent group” for a glance at why this is useful. there must be a less clumsy way to do it

Additionally, there is a key theorem whose usual proof considers cosets (Lagrange’s theorem) which strongly restricts the possible sizes of subgroups of $$G$$, and which itself is enough to classify all the groups of order $$p$$ for $$p$$ prime. Lagrange’s theorem also has very common applications in number theory, in the form of the Fermat-Euler theorem.

Children:

Parents:

• Group

The algebraic structure that captures symmetry, relationships between transformations, and part of what multiplication and addition have in common.