Giovanni Frattini was an Italian

Italy

Italy , officially the Italian Republic languages]] under the European Charter for Regional or Minority Languages. In each of these, Italy's official name is as follows:;;;;;;;;), is a unitary parliamentary republic in South-Central Europe. To the north it borders France, Switzerland, Austria and...

 mathematician

Mathematician

A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

, noted for his contributions to group theory

Group theory

In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...

.

He entered the University of Rome

University of Rome La Sapienza

The Sapienza University of Rome, officially Sapienza – Università di Roma, formerly known as Università degli studi di Roma "La Sapienza", is a coeducational, autonomous state university in Rome, Italy...

  in 1869, where he studied mathematics with Giuseppe Battaglini, Eugenio Beltrami

Eugenio Beltrami

Eugenio Beltrami was an Italian mathematician notable for his work concerning differential geometry and mathematical physics...

, and Luigi Cremona

Luigi Cremona

Luigi Cremona was an Italian mathematician. His life was devoted to the study of geometry and reforming advanced mathematical teaching in Italy. His reputation mainly rests on his Introduzione ad una teoria geometrica delle curve piane...

, obtaining his PhD. in 1875.

In 1885 he published a paper where he defined a certain subgroup

Subgroup

In group theory, given a group G under a binary operation *, a subset H of G is called a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...

 of a finite group

Finite group

In mathematics and abstract algebra, a finite group is a group whose underlying set G has finitely many elements. During the twentieth century, mathematicians investigated certain aspects of the theory of finite groups in great depth, especially the local theory of finite groups, and the theory of...

. This subgroup, now known as the Frattini subgroup

Frattini subgroup

In mathematics, the Frattini subgroup Φ of a group G is the intersection of all maximal subgroups of G. For the case that G is the trivial group e, which has no maximal subgroups, it is defined by Φ = e...

, is the subgroup Φ(G) generated by all the non-generators of the group G. He showed that Φ(G) is nilpotent

Nilpotent group

In mathematics, more specifically in the field of group theory, a nilpotent group is a group that is "almost abelian". This idea is motivated by the fact that nilpotent groups are solvable, and for finite nilpotent groups, two elements having relatively prime orders must commute...

 and, in so doing, developed a method of proof known today as Frattini's argument

Frattini's argument

In group theory, a branch of mathematics, Frattini's argument is an important lemma in the structure theory of finite groups. It is named after Giovanni Frattini, who first used it in a paper from 1885 when defining the Frattini subgroup of a group....

.

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