Girish Mahajan (Editor)

Frattini's argument

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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.

Contents

Statement and proof

Frattini's Argument. If G is a finite group with normal subgroup H, and if P is a Sylow p-subgroup of H, then

G = NG(P)H,NGPPGNGPH

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 ∈ NG(P),

and therefore g ∈ NG(P)H. But g ∈ G was arbitrary, so G = HNG(P) = NG(P)H. ◻

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 NG(NG(P)), it can be shown that NG(NG(P)) = NG(P) whenever G is a finite group and P is a Sylow p-subgroup of G.
  • More generally, if a subgroup M ≤ G contains NG(P) for some Sylow p-subgroup P of G, then M is self-normalizing, i.e. M = NG(M).

    • References

    Frattini's argument Wikipedia
    (Text) CC BY-SA


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