False statementConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-13

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The composition factors determine a finite group up to isomorphism

Statement

False. The composition factors determine a finite group up to isomorphism.

Facts & Assumptions

Given: The cyclic group $C_{4}$ and the direct product $C_{2} \times C_{2}$ .

[F1]

Composition factors are the simple quotients in a composition series (Composition series, composition factors, and composition length).

[L1]

Any two composition series of the same group have equal length and factors agreeing up to isomorphism and permutation (The Jordan–Hölder theorem for groups).

[L2]

Direct products use coordinatewise multiplication ( $G \times H$ is a group with identity $(e_{G}, e_{H})$ , coordinatewise inverses, and homomorphic coordinate projections).

[L3]

A cyclic group generated by an element of finite order $n$ is isomorphic to $Z / n Z$ (Every cyclic group is isomorphic to $(Z, +)$ or to $(Z / n, +)$ for its finite order $n \geq 1$ ).

Refutation

technique · direct

1.1

The chain $C_{4} > C_{2} > 1$ is a composition series with factors $C_{2}, C_{2}$ , by [F1] and [L3].

F1L3

1.2

The chain $C_{2} \times C_{2} > C_{2} \times 1 > 1$ is also a composition series with factors $C_{2}, C_{2}$ , by [F1] and [L2].

F1L2

1.3

The group $C_{4}$ has an element of order four, while every nonidentity element of $C_{2} \times C_{2}$ has order two by coordinatewise multiplication; hence the groups are not isomorphic.

givenL2L3

2.1

Thus two nonisomorphic finite groups have the same composition factors, refuting the statement without contradicting [L1], which compares two series of one group rather than different groups.

step 1.1step 1.2step 1.3L1∎

Depends on

Used by

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Sources

J. S. Milne, Group Theory, Chapter 6 (standard reference, not scraped)
K. Conrad, Subgroup Series I (standard reference, not scraped)
K. Igusa, Notes on Jordan-Hölder, section 5 (standard reference, not scraped)