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Two composition series of $C_{12}$ have the same factors in different orders

Example

Let $C_{12} = ⟨ x ⟩$ . The chains $C_{12} > ⟨ x^{2} ⟩ > ⟨ x^{4} ⟩ > 1$ and $C_{12} > ⟨ x^{3} ⟩ > ⟨ x^{6} ⟩ > 1$ are composition series. Their factor orders are respectively $(2, 2, 3)$ and $(3, 2, 2)$ , so they display the same composition factors in different orders.

Facts & Assumptions

Given: The cyclic group $C_{12} = ⟨ x ⟩$ .

[L1]

Every subgroup of a cyclic group $G = ⟨ g ⟩$ is cyclic; a nontrivial subgroup is generated by the least positive power of $g$ that it contains (Every subgroup of a cyclic group is cyclic; the least positive exponent in a nontrivial subgroup supplies a generator). A quotient of $G = ⟨ g ⟩$ is generated by the image of $g$ and so is cyclic as well.

[L2]

Composition factors are invariant up to isomorphism and permutation (The Jordan–Hölder theorem for groups).

[L3]

The order of a finite group is the product of the orders of the factors in a composition series (The order of a finite group is the product of the orders of its composition factors).

Verification

technique · direct

1.1

Listing the powers of $x$ shows that $⟨ x^{2} ⟩$ , $⟨ x^{4} ⟩$ , $⟨ x^{3} ⟩$ , and $⟨ x^{6} ⟩$ have orders $6, 3, 4, 2$ , respectively; [L1] confirms that all displayed terms are cyclic subgroups.

givenL1algebra

2.1

Each adjacent quotient therefore has prime order: the first list is $2, 2, 3$ , and the second is $3, 2, 2$ . A group of prime order is simple, so both chains are composition series.

step 1.1algebra

3.1

Each quotient is cyclic by [L1], hence the two factor lists are $(C_{2}, C_{2}, C_{3})$ and $(C_{3}, C_{2}, C_{2})$ . Their products both equal $12$ as [L3] requires, and their agreement up to permutation illustrates [L2].

step 1.1step 2.1L1L2L3∎

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)

Two composition series of C12C12​ have the same factors in different orders