**Heinz Prüfer** was a German mathematician who proved important results about abelian groups.

- The town was not an ancient one, being only founded in 1853 and given its present name in 1869 (named for Wilhelm I), twenty-seven years before Heinz was born there.
- Heinz was brought up in a household where science was an everyday part of life.
- After leaving Wilhelmshaven, Prüfer studied at the Gymnasium in Berlin-Zehlendorf completing his schooling there in 1915.
- Now Schur was not at Berlin when Prüfer began his studies but returned to the University of Berlin, taking up a position there in 1916 after holding a professorship in Bonn.
- Frobenius taught Prüfer about Dedekind's way of thinking and his approach to mathematics and this made Prüfer enthusiastic about abstract algebra.
- Prüfer himself later said what a strong impression Schwarz's personality had made on him.
- Prüfer applied himself to his academic studies with extreme diligence, even thinking deeply about comments which others might have passed over as unimportant trifles.
- This way of working would be a characteristic of Prüfer's approach throughout his life as was his independence of thought.
- For his military service, Prüfer was drafted to work at the Königlich Preussische Landesaufnahme in 1917-18.
- It had been Schur's mathematics which had attracted Prüfer most so, after completing his first degree, he undertook research for his doctorate under Schur's supervision.
- Early in his years as a research student working on algebra, Prüfer published his first paper Neuer Beweis eines Satzes über Permutationen Ⓣ(New proof of a theorem on permutations).
- For his doctorate, Prüfer was examined on 28 April 1921.
- The mathematics oral was conducted by Issai Schur and Erhard Schmidt who, respectively, evaluated Prüfer as very good and as excellent.
- Prüfer wrote his doctoral thesis on abelian groups, the topic for which he is best remembered today.
- It is worth noting that the results that Prüfer proved in this work are all stated in terms of countable abelian groups although they hold for abelian groups of arbitrary cardinality.
- After the award of his doctorate, Prüfer was appointed as an assistant in Hamburg.
- However, Prüfer remained at Jena, where he had obtained his permission to teach, and gave courses during the two semesters of session 1926-27.
- Prüfer was an algebraist whose name is familiar to many people even though his bibliography comprises only a few items, four of which are concerned with the structure of abelian groups.
- The posthumous book, Projective Geometry (1935), was edited by G Fleddermann and G Köthe based on Prüfer's lecture notes at Münster University.
- With later editions of the book included, this means that in total we are able to list 15 publications by Prüfer.
- In his next major contribution to abelian groups Prüfer published a two part paper Theorie der Abelschen Gruppen Ⓣ(Theory of Abelian groups).
- In this paper Prüfer stresses that all the results hold for modules over a principal ideal domain.
- The 'Prüfer topology' in introduced in the second paper as is a concept which Lefschetz called 'linearly compact groups' in a paper he published in 1942.
- Prüfer gives a decomposition theorem for 'linearly compact groups' and shows that those of finite rank are the direct product of rank 1 groups.
- In addition to his work on abelian groups, Prüfer also worked on algebraic numbers, publishing the paper Neue Begründung der algebraischen Zahlentheorie Ⓣ(New justifications in algebraic number theory) in 1925, and knot theory.
- In it Prüfer gives a very simple proof of an expansion theorem for a particular second order linear homogeneous differential equation coming from the oscillation and evolution theorem.
- Ludwig Bieberbach and Erich Kamke (1890-1961) both included Prüfer's ideas into the 1930 editions of their respective books.
- After Prüfer's death his lectures notes on projective geometry were collected and published as the 314 page book Projektive Geometrie Ⓣ(Projective geometry) (1933).
- They did, however, inspire a flurry of research by Pietrkowski, Krull, Schöneborn, Leptin and Kaplansky, and linearly compact modules and rings are still objects of active research although few will know that the concept originated with Heinz Prüfer.

Born 10 November 1896, Wilhelmshaven, Germany. Died 7 April 1934, Münster, Germany.

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Origin Germany

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive