Finite not implies composition factor-unique
This article gives the statement and possibly, proof, of a non-implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., finite group) need not satisfy the second group property (i.e., composition factor-unique group)
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The order of the composition factors for different composition series of a finite group need not be the same. In other words, it is possible to have two composition series:
such that there is some for which is not isomorphic to .
Example of a direct product of simple groups
Let be non-isomorphic simple groups. Then, the group has the two composition series:
The quotients for the two series come in different orders: in the first series. For the first series, the quotient is and the quotient is . For the second series, the quotient is and the quotient is . Since is not isomorphic to , the quotients do not occur in the same order.