Group theory and generalizations Example

The alternating group A3 is normal in S3

Builds on Alternating group Symmetric group Index of a subgroup

Example

In the symmetric group $S_3$ , let $A_3 = \{e, (1\,2\,3), (1\,3\,2)\}$ . This is the alternating group of degree $3$ . $A_3$ has $3$ elements and $S_3$ has $6$ , so $A_3$ has index $2$ . The two left cosets are $A_3$ and the set of the three transpositions. The two right cosets are the same two sets. Thus $gA_3 = A_3g$ for each $g \in S_3$ , and $A_3$ is normal in $S_3$ .

Remarks

This shows the result that subgroups of index two are normal.

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

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Alternating group concept
Group concept
Symmetric group concept
Subgroup definition
Left coset definition
Index of a subgroup definition
The alternating group A3 is normal in S3 example

Tag	`T4PCQ68C5K`
URI	`m2://T4PCQ68C5K`
Kind	example
Writing	W0 stub
MSC	20B30
Instance of	Normal subgroup
Illustrates	Subgroups of index two are normal
Uses	Alternating group, Symmetric group, Index of a subgroup

The alternating group A3 is normal in S3

Example

Remarks

Reading path

Further reading

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