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)
View a complete list of group property non-implications | View a complete list of group property implications
Get more facts about finite group|Get more facts about composition factor-unique group

Statement

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:

$\{e\}=M_{0}\leq M_{1}\leq \dots \leq M_{r}=G$

and

$\{e\}=N_{0}\leq N_{1}\leq \dots \leq N_{r}=G$.

such that there is some $i$ for which $N_{i}/N_{i-1}$ is not isomorphic to $M_{i}/M_{i-1}$.

Proof

Example of a direct product of simple groups

Let $G_{1},G_{2}$ be non-isomorphic simple groups. Then, the group $G=G_{1}\times G_{2}$ has the two composition series:

$M_{0}=\{e\}\leq M_{1}=G_{1}\times \{e\}\leq M_{2}=G_{1}\times G_{2}$

and:

$N_{0}=\{e\}\leq N_{1}=\{e\}\times G_{2}\leq N_{2}=G_{1}\times G_{2}$.

The quotients for the two series come in different orders: in the first series. For the first series, the quotient $M_{1}/M_{0}$ is $G_{1}$ and the quotient $M_{2}/M_{1}$ is $G_{2}$. For the second series, the quotient $N_{1}/N_{0}$ is $G_{2}$ and the quotient $N_{2}/N_{1}$ is $G_{1}$. Since $G_{1}$ is not isomorphic to $G_{2}$, the quotients do not occur in the same order.