# N-connected

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.

## Statement and proof

Frattini's Argument.IfGis a finite group with normal subgroupH, and ifPis a Sylowp-subgroup ofH, then

G=N_{G}(P)H,where

N_{G}(P) denotes the normalizer ofPinG.

**Proof**: *P* is a Sylow *p*-subgroup of *H*, so every Sylow *p*-subgroup of *H* is an *H*-conjugate *h*^{−1}*Ph* for some *h* ∈ *H* (see Sylow theorems). Let *g* be any element of *G*. Since *H* is normal in *G*, the subgroup *g*^{−1}*Pg* is contained in *H*. This means that *g*^{−1}*Pg* 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*^{−1}*Pg*=*h*^{−1}*Ph*,

so

*hg*^{−1}*Pgh*^{−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 is a direct product 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*).

## References

- 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 (See Chapter 10, especially Section 10.4.)