Finite not implies composition factor-unique

From Groupprops
Jump to: navigation, search
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 \le M_1 \le \dots \le M_r = G

and

\{ e \} = N_0 \le N_1 \le \dots \le 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 \} \le M_1 = G_1 \times \{ e \} \le M_2 = G_1 \times G_2

and:

N_0 = \{ e \} \le N_1 = \{ e \} \times G_2 \le 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.