group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ( ...

, a branch of

mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, Frattini's argument is an important lemma in the structure theory of

finite group In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving tra ...

s. It is named after Giovanni Frattini, who used it in a paper from 1885 when defining the

Frattini subgroup In mathematics, particularly in group theory, the Frattini subgroup \Phi(G) of a group is the intersection of all maximal subgroups of . For the case that has no maximal subgroups, for example the trivial group or a Prüfer group, it is def ...

of a group. The argument was taken by Frattini, as he himself admits, from a paper of Alfredo Capelli dated 1884.M. Brescia, F. de Giovanni, M. Trombetti
"The True Story Behind Frattini’s Argument"
'' Advances in Group Theory and Applications'' 3
doi:10.4399/97888255036928
/ref>

Frattini's argument

Statement

G

is a finite group with 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 In mathematics, especially group theory, the centralizer (also called commutant) of a subset ''S'' in a group ''G'' is the set \operatorname_G(S) of elements of ''G'' that commute with every element of ''S'', or equivalently, the set of ele ...

P

G

, and

N_G(P)H

means the

product of group subsets In mathematics, one can define a product of group subsets in a natural way. If ''S'' and ''T'' are subsets of a group (mathematics), group ''G'', then their product is the subset of ''G'' defined by :ST = \. The subsets ''S'' and ''T'' need not be s ...

Proof

The group

P

is a Sylow

p

-subgroup of

H

, so every Sylow

p

-subgroup of

H

is an

H

-conjugate of

P

, that is, it is of the form

h^Ph

for some

h \in H

(see

Sylow theorems In mathematics, specifically in the field of finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician Peter Ludwig Sylow that give detailed information about the number of subgroups of fixed ...

). Let

g

be any element of

G

. Since

H

is normal in

G

, the subgroup

g^Pg

is contained in

H

. This means that

g^Pg

is a Sylow

p

-subgroup of

H

. Then, by the above, it must be

H

-conjugate to

P

: that is, for some

h \in H

g^Pg = h^Ph,

and so :

hg^Pgh^ = P.

Thus :

gh^ \in N_G(P),

and therefore

g \in N_G(P)H

. But

g \in G

was arbitrary, and so

G = HN_G(P) = N_G(P)H.\ \square

Applications

* Frattini's argument can be used as part of a proof that any finite

nilpotent group In mathematics, specifically group theory, a nilpotent group ''G'' is a group that has an upper central series that terminates with ''G''. Equivalently, it has a central series of finite length or its lower central series terminates with . I ...

is a

direct product In mathematics, a direct product of objects already known can often be defined by giving a new one. That induces a structure on the Cartesian product of the underlying sets from that of the contributing objects. The categorical product is an abs ...

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  \leq G

contains

N_G(P)

for some Sylow

p

-subgroup

P

G

, then

M

is self-normalizing, i.e.

M = N_G(M)

External links

Frattini's Argument on ProofWiki

Frattini's argument

Statement

Proof

Applications

External links

References

Further reading