Group theory and generalizations Abelian groups Theorem

Ulm's theorem

Builds on Abelian group Prime number Countable set Ordinal number Finite field Dimension Divisible group

Statement

Let $p$ be a prime. An abelian $p$ -group is an abelian group in which every element has order a power of $p$ . It is reduced if its only divisible subgroup is $\set{0}$ . For an abelian $p$ -group $A$ , define subgroups $p^\alpha A$ for each ordinal $\alpha$ by $p^0 A = A$ , $p^{\alpha+1} A = p(p^\alpha A)$ , and $p^\lambda A = \bigcap_{\alpha < \lambda} p^\alpha A$ for limit ordinals $\lambda$ . Let $P_\alpha = \set{x \in p^\alpha A : px = 0}$ , a vector space over the finite field $\F_p$ . The $\alpha$ -th Ulm invariant of $A$ is $u_\alpha(A) = \dim_{\F_p} (P_\alpha / P_{\alpha+1})$ .

Let $A$ and $B$ be countable reduced abelian $p$ -groups. Then $A \cong B$ if and only if $u_\alpha(A) = u_\alpha(B)$ for every ordinal $\alpha$ .

Remarks

Isomorphic groups have the same Ulm invariants, since an isomorphism carries each $P_\alpha$ onto the corresponding subgroup. For the converse one builds an isomorphism step by step, by a back-and-forth argument over the countably many elements. At each step one keeps a partial isomorphism between finite subgroups that preserves the heights of elements. The equality of the Ulm invariants is what makes each extension step possible.

Example: for $A = \Z/p^k\Z$ with $k \ge 1$ , $P_n$ is cyclic of order $p$ for $n < k$ and zero for $n \ge k$ , so $u_{k-1}(A) = 1$ and all other invariants vanish. For a finite abelian $p$ -group, $u_n$ counts the cyclic summands of order $p^{n+1}$ , and the theorem reduces to the classification of finite abelian $p$ -groups.

Countability is necessary. Let $B = \bigoplus_{n \ge 1} \Z/p^n\Z$ , and let $\bar B$ be the torsion part of $\prod_{n \ge 1} \Z/p^n\Z$ . Both are reduced, both have $u_n = 1$ for every finite $n$ and $u_\alpha = 0$ for $\alpha \ge \omega$ , but $B$ is countable and $\bar B$ is uncountable, so they are not isomorphic.

Going deeper

For a student

Ulm's theorem extends the classification of finite abelian $p$ -groups by cyclic decompositions to all countable reduced $p$ -groups, where no cyclic decomposition need exist. A general countable abelian $p$ -group is the direct sum of a divisible group and a reduced group. The divisible part is a direct sum of Prüfer groups, determined by the number of summands. So all countable abelian $p$ -groups are classified. The uncountable case has no such classification, as the example above shows.

Reading path

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Abelian group concept
Countable set concept
Dimension concept
Divisible group concept
Finite field concept
Ordinal number concept
Prime number concept
Ulm's theorem statement

Tag	`D8A4VEQR98`
URI	`m2://D8A4VEQR98`
Kind	statement, theorem
Writing	W0 stub
MSC	20K10
Status	This copy of the site has no build data. Run `m2 status D8A4VEQR98`.
About	Abelian group
Uses	Abelian group, Prime number, Countable set, Ordinal number, Finite field, Dimension, Divisible group
Changed	2 November 2026