ExampleConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-24

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The abelian groups of order $360$ in elementary-divisor and invariant-factor form

Example

Since $360 = 2^{3} \cdot 3^{2} \cdot 5$ , the elementary-divisor and invariant-factor forms pair as follows:

Elementary-divisor form	Invariant-factor form
$Z / 8 \oplus Z / 9 \oplus Z / 5$	$Z / 360$
$Z / 8 \oplus Z / 3 \oplus Z / 3 \oplus Z / 5$	$Z / 3 \oplus Z / 120$
$Z / 4 \oplus Z / 2 \oplus Z / 9 \oplus Z / 5$	$Z / 2 \oplus Z / 180$
$Z / 4 \oplus Z / 2 \oplus Z / 3 \oplus Z / 3 \oplus Z / 5$	$Z / 6 \oplus Z / 60$
$Z / 2 \oplus Z / 2 \oplus Z / 2 \oplus Z / 9 \oplus Z / 5$	$Z / 2 \oplus Z / 2 \oplus Z / 90$
$Z / 2 \oplus Z / 2 \oplus Z / 2 \oplus Z / 3 \oplus Z / 3 \oplus Z / 5$	$Z / 2 \oplus Z / 6 \oplus Z / 30$

Facts & Assumptions

Given: Partitions of positive integers (Partitions of a positive integer).

[L1]

For a finite abelian group, the PID-module elementary divisors and invariant factors agree with the published group-theoretic data (The PID-module and finite-abelian-group classifications have the same canonical data).

Verification

technique · direct

1.1

The exponent $3$ of $2$ has partitions $3$ , $2 + 1$ , and $1 + 1 + 1$ ; the exponent $2$ of $3$ has partitions $2$ and $1 + 1$ ; the exponent $1$ of $5$ has partition $1$ . By [L1], independently combining these choices gives exactly the elementary-divisor rows displayed.

L1given

2.1

Align prime-power factors on the right and multiply columns. The resulting columns are respectively $(360)$ , $(3, 120)$ , $(2, 180)$ , $(6, 60)$ , $(2, 2, 90)$ , and $(2, 6, 30)$ , giving the displayed invariant forms by [L1].

step 1.1L1

3.1

In every row the product of the invariant factors is $360$ , and each factor divides the next. Factoring those factors back into prime powers reproduces its elementary-divisor row, so no regrouping is duplicated.

step 2.1algebra∎

Depends on

Used by

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Dependency tree · two levels

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Sources

K. Conrad, Modules over a PID, finite abelian group specialization (standard reference, not scraped)
M. Brussel, Finitely Generated Modules over a PID, Section 4 (standard reference, not scraped)

The abelian groups of order 360360 in elementary-divisor and invariant-factor form