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The Schreier refinement theorem

Statement

Any two finite subnormal series of a group have equivalent refinements (Subnormal and normal series, factors, refinements, and equivalence).

Facts & Assumptions

Given: Subnormal series $G = G_{0} ⊵ \dots ⊵ G_{m} = 1$ and $G = H_{0} ⊵ \dots ⊵ H_{n} = 1$ .

[F1]

A refinement inserts subgroup terms, and two series are equivalent when their nontrivial factors can be paired up to isomorphism after repetitions are deleted (Subnormal and normal series, factors, refinements, and equivalence).

[L1]

For $A ⊴ A^{*}$ and $B ⊴ B^{*}$ , the butterfly constructions give normal adjacent terms and isomorphic quotient factors (The Zassenhaus butterfly lemma).

Proof

technique · direct

1.1

For $0 \leq i < m$ and $0 \leq j \leq n$ , set $G_{i, j} : = G_{i + 1} (G_{i} \cap H_{j})$ . Then $G_{i, 0} = G_{i}$ and $G_{i, n} = G_{i + 1}$ ; [L1] applied to $G_{i + 1} ⊴ G_{i}$ and $H_{j + 1} ⊴ H_{j}$ gives $G_{i, j + 1} ⊴ G_{i, j}$ .

givenL1

1.2

For $0 \leq j < n$ and $0 \leq i \leq m$ , set $H_{j, i} : = (G_{i} \cap H_{j}) H_{j + 1}$ . Concatenating these chains gives a subnormal refinement of the $H$ -series.

givenL1F1

2.1

Concatenating the finite chains $G_{i, 0} ⊵ \dots ⊵ G_{i, n}$ over $i = 0, \dots, m - 1$ yields a subnormal refinement of the $G$ -series, possibly with repeated adjacent terms.

step 1.1F1

2.2

For every cell $(i, j)$ , [L1] identifies $G_{i, j} / G_{i, j + 1}$ with $H_{j, i} / H_{j, i + 1}$ . Thus the two refinements have their $m n$ displayed factors paired by $(i, j) \leftrightarrow (j, i)$ .

step 1.1step 1.2L1

3.1

Deleting repeated adjacent terms deletes exactly the trivial factors on both sides of each paired cell, so the remaining factors are still paired and isomorphic; the refinements are equivalent.

step 2.1step 2.2F1∎

Depends on

Used by

Dependency tree · two levels

6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

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)