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.

Ҳоло шумо Ҳеҷ ҳисобкунак инро ҳал намекунад, лекин қисмҳои он ҳисобшавандаанд. Яке аз инҳоро дар поён санҷед ё худи худро ворид кунед.

Кор кардан

Дар ҳисоби ройгон ба ҳар як дарс қайдҳо илова карда мешаванд, қайди он, ки шумо чӣ кор кардаед, масъалаҳои ҳалшуда дар як ҷо ва муаллиме, ки шумо метавонед дар бораи ин саҳифа пурсед. Математика барои ҳама кушода аст, хоҳ ворид шуда бошад ё на.

Бақайдгирӣ Ворид шудан

Символы, используемые здесь

Барои муайянкунии пурраи маъно, тасвир ва маънои ҳар як ҳарф дар он, ба ҳар як аломат пахш кунед.

Саволҳои маъмул

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.

Қисми ин саҳифа аз Judson, Abstract Algebra: Theory and Applications (GFDL 1.3). Дар ин ҷо барои мо чизе нест, ки мо ба он бирасем.

& Иловаи забон 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