Lagrange's theorem in S3

Lagrange's theorem in S3
Tag	`4DWC4XFS34`
URI	`m2://4DWC4XFS34`
Kind	example
Writing	W0 stub
MSC	20D60
Illustrates	Lagrange's theorem
Uses	Symmetric group, Subgroup

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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 $6$ .

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Sources

Web https://en.wikipedia.org/wiki/Dihedral_group_of_order_6, Subgroups

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