This is an extended version of review Zbl1446.20003 from zbMATH Open, withthe kind permission of the publishers.Throughout this review, the word ‘group’ denotes abelian group. Althoughexamples and properties of finitely generated groups appear in the works of Galois,Cauchy and Abel on the theory of equations, and of Gauss on quadratic forms andnumber theory, the study of groups as structures in their own right began in the1920s with the analysis of countable torsion groups by Pr¨ufer. In the 1930s, Ulmdiscovered invariants of p-groups which now bear his name. Every p-group G hasa decreasing chain of subgroups pκG, where p0G = G and pκG = p(pκ?1G) if κis a successor ordinal and T α<κ pαG if κ is a limit ordinal. This chain eventuallystabilises at κ = λ, where pλG is the maximum divisible subgroup of G. TheUlm invariant of G is the function u from ordinals to cardinals defined by u(κ) =the dimension of the Z(p)–vector space (pκG)[p]/(pκ+1G)[p], so that u(λ) = 0. In1933, Ulm showed that these invariants classify countable p–groups, while in 1935Zippin showed exactly which functions u arise as the Ulm invariants of countablep-groups.
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