**Giovanni Frattini** attended school in Rome and, completing his schooling in 1869, entered the University of Rome in November of that year to study mathematics. 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. He taught at the Technical Institute there, becoming Head of Mathematics and Descriptive Geometry in the year following his appointment.

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. These are not two of the three papers which have made him famous, the latter being three papers on the generators of finite groups one of which he published in 1885 and the remaining two in 1886.

In the first of these papers *Intorno alla generazione dei gruppi di operazioni* Frattini defined the subgroup which today is known as the Frattini subgroup. His definition was as the subgroup generated by all the non-generators of the group (elements which if included in a generating set for the group can always be omitted to still leave a generating set). He showed that the Frattini subgroup is nilpotent and, in so doing, used the beautiful method of proof known today as the "Frattini argument".

The quality of this work led to Frattini being offered a university chair in Naples, but he declined the offer not wishing to leave Rome for family reasons. Although he was offered a lectureship in algebra at the University of Rome in 1914 he never took up the appointment. By that time he was 62 years of age.

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. His work on differential geometry is important as is his papers on the analysis of second degree indeterminates. On this latter topic he simplified the classical work by Euler, Lagrange and Gauss (anyone would be proud to improve on the work of these three mathematicians!).

The First World War was a difficult time for Frattini who found the events very troubling. In 1915 Italy turned from the Triple Alliance to make a treaty with Britain, France, and Russia. Italy declared war on Austria-Hungary in May of 1915 and some rather indecisive battles were fought in the north. Many Italian soldiers were killed, 500,000 in 1916 alone. Frattini's son was wounded in the war and, in order to be able to support him in the difficult economic conditions, Frattini continued working at a time when he wished to retire. His wife died and poor Frattini suffered a number of unhappy years in his old age.

A glimpse of Frattini's character can be gained by looking at one of his eccentricities. He was a great admirer of the poems of Giuseppe Gioacchino Belli, a Roman poet who wrote around 2,000 sonnets in the Roman dialect. Belli's sonnets:-

... ... express his revolt against literary tradition, the academic mentality, and the social injustices of the papal system. The ritualism of the church and the accepted principles of commonplace morality were also objects of his derision.

Frattini recited the sonnets to his pupils in the Roman dialect in which Belli wrote them. This was reported to the authorities in Rome and a ministerial enquiry was set up. However [1]:-

[

Frattini]declared himself willing to replace these sonnets by the poetry of a highest undersecretary of education. Without further ado, he was allowed to continue his recitation of Belli's poetry.

One can feel from this episode that Frattini was probably an outstanding teacher and indeed this was the case. His belief was that to learn mathematics a student had to do more and read less. I [EFR] absolutely agree with Frattini that one can only learn mathematics by doing mathematics, not by reading about how to do mathematics. Despite Frattini's belief that one should do mathematics rather than read mathematics, he did write a number of excellent books. These do indeed present mathematics in a concrete way [1]:-

... bereft of superfluous abstraction and rich in elegant brilliance.

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.

**Article by:** *J J O'Connor* and *E F Robertson*

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