maths.free › Abstract Algebra › Cosets and Lagrange's Theorem › Cosets

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.

Núna ūú Engin reiknivél leysir þetta, en hlutar þess eru reiknanlegir. Prófaðu einn hér fyrir neðan, eða sláðu inn þinn eigin.

Haltu eigin vinnu þinni

Ókeypis reikningur bætir við athugasemdum við hverja kennslustund, skrá yfir það sem þú hefur lokið, leyst vandamál á einum stað og kennari sem þú getur spurt um þessa síðu.Stærðfræðin sjálf er opin öllum, skráður inn eða ekki.

Skráðu þig inn Innskráning

Tákn notuð hér

Pikkaðu á hvaða tákn sem er fyrir fulla skilgreiningu, mynd og hvað hvert bréf í því merkir.

Spurningar sem fólk spyr

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.

Hlutar þessarar síðu eru aðlagað frá Judson, Abstract Algebra: Theory and Applications (GFDL 1.3). Þjappað og endurútskýrt hér; villur eru okkar.

Meira í Abstract Algebra

Groups Subgroups, cosets and Lagrange's theorem Cyclic groups and permutation groups Homomorphisms, normal subgroups and quotient groups Rings and fields Galois theory: why the quintic has no formula