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

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.