**Giovanni Frattini** was an Italian mathematician who made important contributions to group theory.

- Frattini was taught by some outstanding mathematicians at the University of Rome, being tutored by the geometers Guiseppe Battaglini, Eugenio Beltrami (who had just published his masterpiece on non-euclidean geometry).
- In 1873 Luigi Cremona arrived in Rome and also taught Frattini who obtained his doctorate in 1875.
- After graduating Frattini went to Caltanissetta in central Sicily where he taught at the Liceo, taking up his appointment in 1876.
- At this school Frattini was the head of mathematics but it was a school he was only to teach in for two years for, in November 1878, he moved to Viterbo in central Italy.
- This was much nearer to home for Frattini for Viterbo is situated at the foot of the Cimini Mountains to the northwest of Rome.
- Frattini's road back to Rome was completed in February 1881 when his request for a transfer to the Technical Institute there accepted.
- In 1884 a Military College was founded in Rome and Frattini lectured there from the time that the College opened.
- It was shortly after joining the College that Frattini published three papers on group theory which today make his name familiar to anyone who has studied the topic.
- The route that Frattini had taken to undertake research in group theory had been to study Camille Jordan's papers on the topic.
- As a result of this study, Frattini published two major papers on transitive groups, the first in 1883 and the second in the following year.
- In the first of these papers Intorno alla generazione dei gruppi di operazioni Ⓣ(On the generation of groups of operations) Frattini defined the subgroup which today is known as the Frattini subgroup.
- He showed that the Frattini subgroup is nilpotent and, in so doing, used the beautiful method of proof known today as the "Frattini argument".
- Before commenting on the final years of Frattini's life it is worth noting that he contributed to other areas of mathematics in addition to group theory.
- The First World War was a difficult time for Frattini who found the events very troubling.
- A glimpse of Frattini's character can be gained by looking at one of his eccentricities.
- Frattini recited the sonnets to his pupils in the Roman dialect in which Belli wrote them.
- One can feel from this episode that Frattini was probably an outstanding teacher and indeed this was the case.
- Despite Frattini's belief that one should do mathematics rather than read mathematics, he did write a number of excellent books.
- Those who have seen the "Frattini argument" will agree that "elegant brilliance" is an apt phrase to describe that part of Frattini's work too.

Born 8 January 1852, Rome (now Italy). Died 21 July 1925, Rome, Italy.

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Algebra, Group Theory, Origin Italy

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