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Cosets

Let G be a group and H a subgroup of G. Define a left coset of H with representative g \in G to be the set \[\begin{aligned}\end{aligned}\]. Right cosets can be defined similarly by \[\begin{aligned}\end{aligned}\].

Cosets

Let \(G\) be a group and \(H\) a subgroup of \(G\). Define a left coset of \(H\) with representative \(g \in G\) to be the set \[\begin{aligned}\end{aligned}\]. Right cosets can be defined similarly by \[\begin{aligned}\end{aligned}\]. If left and right cosets coincide or if it is clear from the context to which type of coset that we are referring, we will use the word coset without specifying left or right.

Example

Let \(H\) be the subgroup of \({\mathbb Z}_6\) consisting of the elements \(0\) and \(3\). The cosets are \[\begin{aligned}0 + H = 3 + H = \{ 0, 3 \} \\ 1 + H = 4 + H = \{ 1, 4 \} \\ 2 + H = 5 + H = \{ 2, 5 \}\end{aligned}\]. We will always write the cosets of subgroups of \({\mathbb Z}\) and \({\mathbb Z}_n\) with the additive notation we have used for cosets here. In a commutative group, left and right cosets are always identical.

Example

Let \(H\) be the subgroup of \(S_3\) defined by the permutations \(\{(1), (1 \, 2 \, 3), (1 \, 3 \, 2) \}\). The left cosets of \(H\) are \[\begin{aligned}(1)H = (1 \, 2 \, 3)H = (1 \, 3 \, 2)H = \{(1), (1 \, 2 \, 3), (1 \, 3 \, 2) \} \\ (1 \, 2)H = (1 \, 3)H = (2 \, 3)H = \{ (1 \, 2), (1 \, 3), (2 \, 3) \}\end{aligned}\]. The right cosets of \(H\) are exactly the same as the left cosets: \[\begin{aligned}H(1) = H(1 \, 2 \, 3) = H(1 \, 3 \, 2) = \{(1), (1 \, 2 \, 3), (1 \, 3 \, 2) \} \\ H(1 \, 2) = H(1 \, 3) = H(2 \, 3) = \{ (1 \, 2), (1 \, 3), (2 \, 3) \}\end{aligned}\].

It is not always the case that a left coset is the same as a right coset. Let \(K\) be the subgroup of \(S_3\) defined by the permutations \(\{(1), (1 \, 2)\}\). Then the left cosets of \(K\) are \[\begin{aligned}(1)K = (1 \, 2)K = \{(1), (1 \, 2)\} \\ (1 \, 3)K = (1 \, 2 \, 3)K = \{(1 \, 3), (1 \, 2 \, 3)\} \\ (2 \, 3)K = (1 \, 3 \, 2)K = \{(2 \, 3), (1 \, 3 \, 2)\};\end{aligned}\] however, the right cosets of \(K\) are \[\begin{aligned}K(1) = K(1 \, 2) = \{(1), (1 \, 2)\} \\ K(1 \, 3) = K(1 \, 3 \, 2) = \{(1 \, 3), (1 \, 3 \, 2)\} \\ K(2 \, 3) = K(1 \, 2 \, 3) = \{(2 \, 3), (1 \, 2 \, 3)\}\end{aligned}\].

The following lemma is quite useful when dealing with cosets. (We leave its proof as an exercise.)

In all of our examples the cosets of a subgroup \(H\) partition the larger group \(G\). The following theorem proclaims that this will always be the case.

Example

Let \(G= {\mathbb Z}_6\) and \(H = \{ 0, 3 \}\). Then \([G:H] = 3\).

Condensed — the full section is in Judson, Abstract Algebra: Theory and Applications.

Symbols used here

x \in A,\ A \subseteq B
element of, subset
x belongs to A; every element of A is in B.
\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}
number sets
Naturals, integers, rationals, reals, complex numbers.
\blacksquare\ \text{or}\ \square
end of proof (halmos)
Marks the point where the statement has been established.
a \equiv b \pmod n
congruent modulo n
n divides a − b; a and b have the same remainder.
a \mid b,\ \gcd(a,b)
divides, greatest common divisor
b is a multiple of a; the largest number dividing both.
(G, \cdot),\ e,\ g^{-1}
group, identity, inverse
A set with an operation; the do-nothing element; the element that undoes g.
G \cong H,\ G / N
isomorphic, quotient group
Same structure; the group of cosets of a normal subgroup N.
\mathbb{Z}/n\mathbb{Z},\ \mathbb{Z}_n
integers modulo n
The remainders 0…n−1 with clock arithmetic.
\operatorname{Hom}(A, B),\ f \circ g
arrows from A to B, composition
The set of morphisms; do g then f.

Questions people ask

What is a group, in plain words?

A set with one operation that is associative, has an identity, and lets every element be undone. Symmetries of any object form a group — that is where the idea came from.

What is the difference between a ring and a field?

A ring has addition and multiplication that behave like the integers (you cannot always divide); a field is a ring where every non-zero element has a reciprocal, like the rationals or the reals.

Δοκίμασε μόνος σου.

Parts of this page are adapted from Judson, Abstract Algebra: Theory and Applications (GFDL 1.3). Condensed and re-explained here; errors are ours.

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