**Alfredo Capelli**attended the University of Rome where, in the academic year 1875-1876, he attended a course by Giuseppe Battaglini on the theory of groups of substitutions. In this course Battaglini followed the approach given by Camille Jordan in his

*Traité des substitutions et des équations algebraique*Ⓣ which he published in 1870. Battaglini [11]:-

In fact Capelli had published two notes on the theory of groups in the same journal two years earlier while he was working on his thesis supervised by Battaglini. These two notes, which were based on material from Serret's... advised the young Capelli to tackle problems on groups in his thesis. From this thesis sprang a huge memoir, appearing in the 'Giornale di matematiche' in1878.

*Cours d'algèbre supérieure*Ⓣ, were entitled

*Dimostrazione di due proprietà numeriche offerte dalla teoria delle sostituzioni permutabili con una sostituzione data*Ⓣ and

*Intorno ai valori di una funzione lineare di più variabili*Ⓣ. His 1878 paper

*Sopra l'isomorfismo dei gruppi di sostituzioni*Ⓣ was a major contribution of 55 pages, essentially publishing the results from his thesis. It is a remarkable contribution which contains a proof of Sylow's theorems and of Jordan's theorem on composition series. These were independent discoveries by Capelli since he was unaware of Sylow's 1872 paper and Netto's 1874 paper where these results were first published. His proofs are interesting since they are quite different from those in Sylow's and Netto's papers; in particular he gives a simpler proof of Sylow's theorem. However, Capelli's paper also contains standard results later proved by Frobenius and Burnside which have been attributed to these later authors. Capelli's paper contains the following interesting observation on isomorphisms:-

The authors of [3] discuss why Capelli was unaware of Sylow's results and why Burnside and other early 20The importance of the concept of isomorphism has already been emphasized and validated through applications, in such a way that, by reading the principles and considering the fertility of Jordan's excellent treatise, I believed I had found in it the natural key for studying substitutions, especially when one does not wish to involve analysis.

^{th}century authors were unaware of Capelli's results about nilpotent groups and composition series.

Capelli graduated from the University of Rome in 1877 and then continued to develop his mathematical skills working as Felice Casorati's assistant at the University of Pavia. He also spent time at the University of Berlin where he was influenced by Karl Weierstrass and Leopold Kronecker. Casorati corresponded regularly with Weierstrass and had developed a strong link between Italian and German mathematicians so Capelli's visit to Berlin was a natural one in this context. Over this period, Capelli did not work on group theory but his interests had moved to the theory of algebraic forms. In 1881 he was appointed as professor of Algebraic Analysis at the University of Palermo [4]:-

In 1884 Capelli published another major work on group theoryIn1881, Capelli obtained the position of professor of algebra at the University of Palermo. There, he found a situation different from that in the rest of the Kingdom. In fact, in the mid-1870s, Sicily was still awaiting a sign of cultural renewal, while elsewhere in Italy, men such as Cremona, Brioschi, and Betti were almost concluding their radical reform of public education at all levels. In Palermo, the situation finally began to change in1878with the arrival of Cesare Arzelà(1847-1912), who held the chair of algebra for two years, and of Capelli, who replaced Arzelà when the latter moved to Bologna.

*Sopra la composizione dei gruppi di sostituzioni*Ⓣ. In this he proved important results on nilpotent groups (although he did not use the term 'nilpotent group' which is modern). In fact he gave what is today known as the 'Frattini argument'. Why, one might ask is it not then known today as the 'Capelli argument'. In fact Capelli and Frattini had both been students of Battaglini, with Frattini graduating two years before Capelli. They had remained close friends sharing a common interest in group theory and communicating their results to each other. In 1885 Frattini published

*Intorno alla generazione dei gruppi di operazioni*Ⓣ and in this paper he defined the subgroup known today as the Frattini subgroup. He also used the beautiful 'Frattini argument', although he clearly stated that this was due to Capelli. Perhaps Frattini should have used the term 'Capelli argument' himself as this would have guaranteed Capelli receiving credit. As it happens the name 'Frattini argument' entered common use.

While at Palermo, Capelli collaborated with Giovanni Garbieri (1847-1931) who had succeeded Giusto Bellavitis at the University of Padua in 1882. The results of their collaboration was the famous book *Corso di analisi algebrica* Ⓣ published in 1886. Capelli had proved the theorem, known today as the Rouché-Capelli theorem, which gives conditions for the existence of the solution of a system of linear equations. In 1879 Frobenius defined the rank of a system of equations to be the maximal order of a nonzero minor. This left the concept of rank where it had long been, namely tied to the theory of determinants. In 1886 Capelli and Garbieri in *Corso di analisi algebrica* Ⓣ showed that a system of equations having rank k is equivalent to a triangular system with exactly k nonzero diagonal terms. Jean-Luc Dorier writes [1]:-

Then they pointed out, and proved, that the rank of the lines of a matrix is the same as the rank of its rows. They also showed that a system of equations is consistent if and only if the rank of the array of its coefficients is the same as the rank of the array augmented by the row of constant terms.In their proof they applied a method of elimination using minors of the system as coefficients but, even if their approach used technical results of the theory of determinants, the new result they introduced pointed out an invariant linked to the rank which had nothing to do with determinants. This approach is very important in effective methods for solving systems of linear equations ...

In 1886, the year this book was published, Capelli entered the competition for the chair of algebra at the University of Naples. Filling chairs by competition was the standard method in Italy at this time. He was appointed and remained in Naples for the rest of his life. He published over 80 papers throughout his career so we can only mention a few examples to illustrate his contributions while in Naples. In *Sulla limitata possibilità di trasformazioni conformi nello spazio* Ⓣ (1886), Capelli gives a geometrical proof of Liouville's theorem on when conformal mappings are Möbius transformations. He published *Über die Zurückführung der Cayley'schen Operation W auf gewöhnliche Polar-Operationen* Ⓣ in 1887, and then, in *Sur les opérations dans la théorie des formes algébraique* Ⓣ (1890), Capelli gives generators for the centre of the enveloping algebra of the Lie algebra *GL*(*n*). In papers in 1887 and 1890, Capelli introduced what became known as the 'Capelli identities' which involved equalities of differential operators. These played a major role in Hermann Weyl's book *The Classical Groups*. In 1897 Capelli, in a paper in the *Giornale di matematiche*, gave the following test for the symmetry of a polynomial:

A necessary and sufficient condition for a polynomial to be symmetric is that it be unchanged by a cyclic interchange of all its variables, as well as by the cyclic interchange of all but one of its variables, the variables having the same relative positions in the two cycles.

Finally we mention Capelli's 1901 paper *Sulla riduttibilità della funzione x ^{n}- A in un campo qualunque di razionalità* Ⓣ.

As well papers, Capelli published a number of books. We have already mentioned his joint work with Giovanni Garbieri written in Palermo and published in 1886. In the years he spent in Palermo, Capelli taught algebraic analysis for engineers. He gathered the content of the courses he had given on the subject in Palermo, and also those given later in Naples, and created the book *Istituzioni di analisi algebrica* Ⓣ (first edition 1894, second edition 1898, third edition 1902). His other books include *Lezioni di algebra complementare ad uso degli aspiranti alla licenza universitaria Napoli* Ⓣ (1895) and *Lezioni sulla teoria delle forme algebriche* Ⓣ (1902).

Capelli continued to hold the chair of algebra in Naples until his death. In 1894, on the death of Battaglini, Capelli took over the editorship of the *Giornale di matematiche* which, from that time on, became known as the *Giornale di Matematiche di Battaglini*. He continued in this role of editor until his death as the result of a heart attack in 1910. He had been honoured with being elected to the Accademia nazionale dei Lincei, the Accademia nazionale dei Lombardo, the Accademia della Scienze di Napoli, and the Accademia della Scienze, Lettere e Belle Arti di Palermo.

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

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