Group theory and generalizations Example

Lagrange's theorem in S3

Example

The symmetric group $S_3$ has order $6$ . Its subgroups are $\{e\}$ , the three subgroups $\{e, (1\,2)\}$ , $\{e, (1\,3)\}$ and $\{e, (2\,3)\}$ , the subgroup $A_3$ of order $3$ , and $S_3$ . Their orders are $1$ , $2$ , $3$ and $6$ . Each of these numbers divides $6$ , as Lagrange's theorem states.

Remarks

No subgroup has order $4$ or $5$ , because these numbers do not divide $6$ .

Reading path

To understand this page from the ground up, read these 3 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

Tag	`4DWC4XFS34`
URI	`m2://4DWC4XFS34`
Kind	example
Writing	W0 stub
MSC	20D60
Illustrates	Lagrange's theorem
Uses	Symmetric group, Subgroup

Lagrange's theorem in S3

Example

Remarks

Reading path

Further reading

What this page needs for W1

What links here