## Professor W Gaschütz - some reminiscences

In the year 2000, when Wolfgang Gaschütz was eighty years old, Roger Carter published 'Professor W Gaschütz - some reminiscences' in

*Proc. F. Scorina Gomel State Univ. No.***3**(16) (2000), 17. We present below a version of that tribute:I would like to recount briefly, in this celebratory issue for Wolfgang Gaschütz, some of my mathematical and personal encounters with him. Professor Gaschütz influenced considerably my work as a young mathematician. I wrote my Ph.D. thesis in 1959 at Cambridge University, working under the supervision of Derek Taunt. The thesis concerned properties of the system normalizers of a finite soluble group, which had been defined by Philip Hall in 1937. After completing the thesis I spent a postdoctoral year 1959-60 at the University of Tübingen to visit Professor H Wielandt and, under the influence of P Hall and H Wielandt, I proved the existence and conjugacy of nilpotent self-normalizing subgroups in a finite soluble group.

In 1960 I took up a lectureship in Newcastle and in 1962 I paid a visit to Stockholm to attend the International Congress of Mathematicians. During this meeting I learned from Joachim Neubüser that Gaschütz had obtained new results on finite soluble groups containing my results as a special case. As the train back from Stockholm to London stopped quite close to Kiel, I decided to take the opportunity to get out and visit Professor Gaschütz. When I arrived at Kiel I found that he was at a family birthday party, but he very kindly arranged to have a discussion with me, and told me about his new theory of formations and covering subgroups. This conversation inspired me to start working on formations and the so-called F-normalizers in a finite soluble group, which generalise the system normalizers which I had studied in my Ph.D. thesis. I subsequently found that Trevor Hawkes had obtained similar results independently, and so we published them in a joint paper, which eventually appeared in 1967.

Although my research interests subsequently moved away from soluble groups I still had the opportunity to meet Gaschütz fairly regularly at group theory meetings at the Mathematical Research Centre at Oberwolfach, where he was frequently accompanied by Frau Gaschütz. I remember vividly a lecture he gave there on automorphisms of p-groups, showing that finite non-abelian p-groups possess outer p-automorphisms. Gaschütz spoke for about 40 minutes only, but with such style and panache that his talk outshone the other more conventional one hour lectures.

I also had the pleasure of welcoming Gaschütz on several occasions as a visitor to the University of Warwick, which I had joined in 1965, to take part in the Warwick Algebra Symposia which were organised there. I always looked forward to his lively expositions of his continuing work on soluble groups. I also enjoyed touring around the Cotswolds with him by car, and seeing his reaction to some of the ancient buildings we visited there.

My most recent meeting with Professor Gaschütz took place in 1987. While on an extended visit to Essen I had the opportunity of travelling to Kiel to visit Gaschütz shortly before his retirement. On this occasion, as on earlier occasions, I was again conscious of the great pleasure I gained from talking to Herr Gaschütz, in particular from his genial and expansive manner.

I send him and his family all good wishes on his 80th birthday.