# example of a Jordan-Hölder decomposition

A group that has a composition series will often have several different composition series.

For example, the cyclic group $C_{12}$ has $(E,C_{2},C_{6},C_{12})$, and $(E,C_{2},C_{4},C_{12})$, and $(E,C_{3},C_{6},C_{12})$ as different composition series. However, the result of the Jordan-Hölder Theorem is that any two composition series of a group are equivalent, in the sense that the sequence of factor groups in each series are the same, up to rearrangement of their order in the sequence $A_{i+1}/A_{i}$. In the above example, the factor groups are isomorphic to $(C_{2},C_{3},C_{2})$, $(C_{2},C_{2},C_{3})$, and $(C_{3},C_{2},C_{2})$, respectively.

This is taken from the http://en.wikipedia.org/wiki/Solvable_groupWikipedia article on solvable groups.

Title example of a Jordan-Hölder decomposition ExampleOfAJordanHolderDecomposition 2013-03-22 14:24:33 2013-03-22 14:24:33 mathcam (2727) mathcam (2727) 10 mathcam (2727) Example msc 20E15 example of Jordan-Holder decomposition