Math 2.0

example

The left cosets of a subgroup of order two in S3

Example

In the symmetric group S3S_3, let H={e,(1 2)}H = \{e, (1\,2)\}. Compose permutations from right to left. The left cosets of HH are eH={e,(1 2)}eH = \{e, (1\,2)\}, (1 3)H={(1 3),(1 2 3)}(1\,3)H = \{(1\,3), (1\,2\,3)\} and (2 3)H={(2 3),(1 3 2)}(2\,3)H = \{(2\,3), (1\,3\,2)\}. Each of the 66 elements of S3S_3 is in exactly one of these cosets. Each coset has 22 elements. Thus 6=3⋅26 = 3 \cdot 2, as Lagrange's theorem states.

Remarks

The right cosets of HH are different sets, because HH is not normal in S3S_3.

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Group, concept, no evidence levelGroupconceptSymmetric group, concept, no evidence levelSymmetric groupconceptSubgroup, definition, no evidence levelSubgroupdefinitionLeft coset, definition, no evidence levelLeft cosetdefinitionThe left cosets of a subgroup of order two in S3, example, no evidence levelThe left cosets of a …example
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  1. Group concept
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  1. Web https://en.wikipedia.org/wiki/Coset, First example

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