In 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...

, a branch of mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, Frattini's argument is an important lemma

Lemma (mathematics)

In mathematics, a lemma is a proven proposition which is used as a stepping stone to a larger result rather than as a statement in-and-of itself...

 in the structure theory of 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...

s. It is named after Giovanni Frattini

Giovanni Frattini

Giovanni Frattini was an Italian mathematician, noted for his contributions to group theory.He entered the University of Rome in 1869, where he studied mathematics with Giuseppe Battaglini, Eugenio Beltrami, and Luigi Cremona, obtaining his PhD. in 1875.In 1885 he published a paper where he...

, who first used it in a paper from 1885 when defining 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...

 of a group.

Statement and proof

Frattini's argument states that if a finite group G has a normal subgroup H, and if P is a Sylow p-subgroup of H, then

G = N_G(P)H,

where N_G(P) denotes the normalizer

Centralizer and normalizer

In group theory, the centralizer and normalizer of a subset S of a group G are subgroups of G which have a restricted action on the elements of S and S as a whole, respectively...

 of P in G.

Proof: P is a Sylow p-subgroup of H, so every Sylow p-subgroup of H is an H-conjugate h⁻¹Ph for some h ∈ H (see Sylow theorems).
Let g be any element of G. Since H is normal in G, the subgroup
g⁻¹Pg is contained in H. This means that
g⁻¹Pg is a Sylow p-subgroup of H. Then by the above, it must be H-conjugate to P: that is, for some h ∈ H

g⁻¹Pg = h⁻¹Ph,

so

hg⁻¹Pgh⁻¹ = P; thus

gh⁻¹ ∈ N_G(P),

and therefore g ∈ N_G(P)H. But g ∈ G was arbitrary, so G = HN_G(P) = N_G(P)H.

Applications

• Frattini's argument can be used as part of a proof that any finite nilpotent group
  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...
  
   is a direct product
  Direct product of groups
  In the mathematical field of group theory, the direct product is an operation that takes two groups and and constructs a new group, usually denoted...
  
  of its Sylow subgroups.
• By applying Frattini's argument to N_G(N_G(P)), it can be shown that N_G(N_G(P)) = N_G(P) whenever G is a finite group and P is a Sylow p-subgroup of G.
• More generally, if a subgroup M ≤ G contains N_G(P) for some Sylow p-subgroup P of G, then M is self-normalizing, i.e. M = N_G(M).

Proof: M is normal in H := N_G(M), and P is a Sylow p-subgroup of M, so the Frattini argument applied to the group H with normal subgroup M and Sylow p-subgroup P gives N_H(P)M = H. Since N_H(P) ≤ N_G(P) ≤ M, one has the chain of inclusions M ≤ H = N_H(P)M ≤ M M = M, so M = H.