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Baer won a scholarship for specially gifted students in 1924 and this enabled him to study at Kiel for a year with Hasse, Steinitz and Toeplitz. During this year in Kiel, Baer wrote up his doctoral dissertation and presented it to Göttingen in 1925. It was published in Crelle's Journal in 1927.
A post at Freiburg was offered to him by Loewy. Baer held this post from 1926 until 1928 and, during this time, he turned towards algebra. Loewy clearly was one of the main influences in this change of direction.
Hasse offered him a post at Halle in 1928 and Baer accepted. While at Halle he undertook a joint project with Hasse, publishing Steinitz's Algebraische Theorie der Körper, which had been first published in Crelle's Journal in 1910, as a book with a commentary on the text and an appendix on Galois theory written by Baer.
While Baer was on holiday in Austria with his wife, Hitler came to power and Baer was informed that his services at Halle were no longer required. He then accepted an invitation from Mordell to go to Manchester, and after going to Oxford to meet Weyl, he accepted his invitation to Princeton.
Baer had begun studying infinite abelian groups while at Manchester and he continued his study of this topic at Princeton where he stayed from 1935 until 1937. He moved to North Carolina but when he was offered a chair at The University of Illinois at Urbana in 1938 he accepted the post. He remained there for until 1956 when he returned to Frankfurt in Germany where Moufang held a senior position.
His mathematical work, some of which has been mentioned above, was wide ranging; topology, abelian groups and geometry. His most important work, however, was in group theory; on the extension problem for groups, finiteness conditions, soluble and nilpotent groups.
In 1940 he introduced the concept of an injective module, then began studying group actions in geometry. He applied group theory to the study of projective planes and his work in this area led to the topic of combinatorics as we know it today.
From 1950 onwards his work turned more towards finiteness conditions on groups and generalisations of soluble and nilpotent groups. Many concepts in this area were introduced by him, in particular the Baer radical of a group and Baer groups (groups in which every cyclic subgroup is subnormal).
I (EFR) heard him lecture at the University of Warwick in 1977. This was a sad occasion as by this time Baer knew that he was seriously ill (with stomach cancer). He described work which he felt he wanted to communicate but felt that he would not live long enough to be able to polish it to his high standards.
An operation in 1978 was successful and Baer enjoyed a while longer creating the mathematics he loved and communicating it to others showing his excitement and joy in his subject. He enjoyed a final Oberwolfach meeting in May 1979 where he was
a trim and fit figure, dressed in an openneck white shirt, grey flannel trousers and tennis shoes. A happy smile on his face... .References (4 books/articles)