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 used it in a paper from 1885 when defining the Frattini subgroup of a group. The argument was taken by Frattini, as he himself admits, from a paper of Alfredo Capelli dated 1884.

Frattini's Argument

Statement

If [math]\displaystyle{ G }[/math] is a finite group with normal subgroup [math]\displaystyle{ H }[/math], and if [math]\displaystyle{ P }[/math] is a Sylow p-subgroup of [math]\displaystyle{ H }[/math], then

[math]\displaystyle{ G = N_{G}(P)H }[/math]

where [math]\displaystyle{ N_{G}(P) }[/math] denotes the normalizer of [math]\displaystyle{ P }[/math] in [math]\displaystyle{ G }[/math] and [math]\displaystyle{ N_{G}(P)H }[/math] means the product of group subsets.

Proof

The group [math]\displaystyle{ P }[/math] is a Sylow [math]\displaystyle{ p }[/math]-subgroup of [math]\displaystyle{ H }[/math], so every Sylow [math]\displaystyle{ p }[/math]-subgroup of [math]\displaystyle{ H }[/math] is an [math]\displaystyle{ H }[/math]-conjugate of [math]\displaystyle{ P }[/math], that is, it is of the form [math]\displaystyle{ h^{-1}Ph }[/math], for some [math]\displaystyle{ h \in H }[/math] (see Sylow theorems). Let [math]\displaystyle{ g }[/math] be any element of [math]\displaystyle{ G }[/math]. Since [math]\displaystyle{ H }[/math] is normal in [math]\displaystyle{ G }[/math], the subgroup [math]\displaystyle{ g^{-1}Pg }[/math] is contained in [math]\displaystyle{ H }[/math]. This means that [math]\displaystyle{ g^{-1}Pg }[/math] is a Sylow [math]\displaystyle{ p }[/math]-subgroup of [math]\displaystyle{ H }[/math]. Then by the above, it must be [math]\displaystyle{ H }[/math]-conjugate to [math]\displaystyle{ P }[/math]: that is, for some [math]\displaystyle{ h \in H }[/math]

[math]\displaystyle{ g^{-1}Pg = h^{-1}Ph }[/math],

and so

[math]\displaystyle{ hg^{-1}Pgh^{-1} = P }[/math].

Thus,

[math]\displaystyle{ gh^{-1} \in N_{G}(P) }[/math],

and therefore [math]\displaystyle{ g \in N_{G}(P)H }[/math]. But [math]\displaystyle{ g \in G }[/math] was arbitrary, and so [math]\displaystyle{ G = HN_G(P) = N_G(P)H. \,\, \square }[/math]

Applications

• Frattini's argument can be used as part of a proof that any finite nilpotent group is a direct product of its Sylow subgroups.
• By applying Frattini's argument to [math]\displaystyle{ N_G(N_G(P)) }[/math], it can be shown that [math]\displaystyle{ N_G(N_G(P)) = N_G(P) }[/math] whenever [math]\displaystyle{ G }[/math] is a finite group and [math]\displaystyle{ P }[/math] is a Sylow [math]\displaystyle{ p }[/math]-subgroup of [math]\displaystyle{ G }[/math].
• More generally, if a subgroup [math]\displaystyle{ M \leq G }[/math] contains [math]\displaystyle{ N_G(P) }[/math] for some Sylow [math]\displaystyle{ p }[/math]-subgroup [math]\displaystyle{ P }[/math] of [math]\displaystyle{ G }[/math], then [math]\displaystyle{ M }[/math] is self-normalizing, i.e. [math]\displaystyle{ M = N_G(M) }[/math].

External links

• Frattini's Argument on ProofWiki

References

• Hall, Marshall (1959). The theory of groups. New York, N.Y.: Macmillan.  (See Chapter 10, especially Section 10.4.)