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The PID-module and finite-abelian-group classifications have the same canonical data
Statement
For a finite abelian group, the PID-module elementary divisors and invariant factors agree with the published group-theoretic data. In particular the two classifications attach the same prime-power multiset and the same divisibility chain, including the empty data for the trivial group.
Facts & Assumptions
Given: The module-derived abelian classification of The fundamental theorem of finitely generated abelian groups from PID modules and the group-side definitions of Elementary-divisor data for a finite abelian group and Invariant-factor data for a finite abelian group.
Every finite abelian group is isomorphic to a finite direct product of cyclic groups of prime-power order, with unique factor orders up to permutation (Fundamental theorem of finite abelian groups: elementary-divisor form).
Every finite abelian group has a unique divisibility list of invariant factors, with the trivial group represented by the empty list (Fundamental theorem of finite abelian groups: invariant-factor form).
Proof
A finite abelian group cannot contain a nonzero free summand , so its module-theoretic free rank is .
The module elementary-divisor form and [L1] both express the group as cyclic groups of prime-power order. Their uniqueness clauses force the same prime powers with the same multiplicities, which is exactly agreement of the two elementary-divisor definitions.
Aligning those common prime powers produces the module invariant factors, while [L2] uniquely characterizes the group invariant-factor chain. The two chains therefore agree. Conversely either common list reconstructs the same cyclic direct sum, proving agreement in both directions.
Depends on
- The fundamental theorem of finitely generated abelian groups from PID modules
- Fundamental theorem of finite abelian groups: elementary-divisor form
- Fundamental theorem of finite abelian groups: invariant-factor form
- Elementary-divisor data for a finite abelian group
- Invariant-factor data for a finite abelian group
Used by
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Sources
- K. Conrad, Modules over a PID, Example 4.2 (standard reference, not scraped)
- M. Brussel, Finitely Generated Modules over a PID, Section 4 (standard reference, not scraped)