Group theory and generalizations Example

The left cosets of a subgroup of order two in S3

Example

In the symmetric group $S_3$ , let $H = \{e, (1\,2)\}$ . Compose permutations from right to left. The left cosets of $H$ are $eH = \{e, (1\,2)\}$ , $(1\,3)H = \{(1\,3), (1\,2\,3)\}$ and $(2\,3)H = \{(2\,3), (1\,3\,2)\}$ . Each of the $6$ elements of $S_3$ is in exactly one of these cosets. Each coset has $2$ elements. Thus $6 = 3 \cdot 2$ , as Lagrange's theorem states.

Remarks

The right cosets of $H$ are different sets, because $H$ is not normal in $S_3$ .

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Group concept
Symmetric group concept
Subgroup definition
Left coset definition
The left cosets of a subgroup of order two in S3 example

Tag	`6VFF2TTXYZ`
URI	`m2://6VFF2TTXYZ`
Kind	example
Writing	W0 stub
MSC	20B30
Instance of	Left coset
Illustrates	Lagrange's theorem
Uses	Symmetric group, Left coset

The left cosets of a subgroup of order two in S3

Example

Remarks

Reading path

Further reading

What this page needs for W1

What links here