Composition factor-unique not implies composition series-unique

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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., composition factor-unique group) need not satisfy the second group property (i.e., composition series-unique group)
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It is possible to have a group (in fact, a finite group) such that any two composition series for the group have the same composition factors occurring in the same order, but such that there is more than one composition series.

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Direct product of isomorphic simple groups

Suppose S is a simple group. Let G := S \times S be the direct product of isomorphic copies of S. Note that we have two composition series:

\{ e \} \le S \times \{ e \} \le S \times S


\{ e \} \le \{ e \} \times S \le S \times S.

Thus, G does not have a unique composition series. On the other hand, any composition series of G has the simple group S occurring twice, so all composition series have the composition factors occurring in the same order.

(Note that when S is simple non-Abelian, these are the only two composition series: this follows from the fact that normal subdirect product of perfect groups equals direct product). When S is simple Abelian, it is a cyclic group of prime order p, and there are p + 1 possible composition series.