• Source: Frattini subgroup

In mathematics, particularly in group theory, the Frattini subgroup



Φ
(
G
)


{\displaystyle \Phi (G)}

of a group G is the intersection of all maximal subgroups of G. For the case that G has no maximal subgroups, for example the trivial group {e} or a Prüfer group, it is defined by



Φ
(
G
)
=
G


{\displaystyle \Phi (G)=G}

. It is analogous to the Jacobson radical in the theory of rings, and intuitively can be thought of as the subgroup of "small elements" (see the "non-generator" characterization below). It is named after Giovanni Frattini, who defined the concept in a paper published in 1885.




Some facts






Φ
(
G
)


{\displaystyle \Phi (G)}

is equal to the set of all non-generators or non-generating elements of G. A non-generating element of G is an element that can always be removed from a generating set; that is, an element a of G such that whenever X is a generating set of G containing a,



X
∖
{
a
}


{\displaystyle X\setminus \{a\}}

is also a generating set of G.




Φ
(
G
)


{\displaystyle \Phi (G)}

is always a characteristic subgroup of G; in particular, it is always a normal subgroup of G.
If G is finite, then



Φ
(
G
)


{\displaystyle \Phi (G)}

is nilpotent.
If G is a finite p-group, then



Φ
(
G
)
=

G

p


[
G
,
G
]


{\displaystyle \Phi (G)=G^{p}[G,G]}

. Thus the Frattini subgroup is the smallest (with respect to inclusion) normal subgroup N such that the quotient group



G

/

N


{\displaystyle G/N}

is an elementary abelian group, i.e., isomorphic to a direct sum of cyclic groups of order p. Moreover, if the quotient group



G

/

Φ
(
G
)


{\displaystyle G/\Phi (G)}

(also called the Frattini quotient of G) has order




p

k




{\displaystyle p^{k}}

, then k is the smallest number of generators for G (that is, the smallest cardinality of a generating set for G). In particular a finite p-group is cyclic if and only if its Frattini quotient is cyclic (of order p). A finite p-group is elementary abelian if and only if its Frattini subgroup is the trivial group,



Φ
(
G
)
=
{
e
}


{\displaystyle \Phi (G)=\{e\}}

.
If H and K are finite, then



Φ
(
H
×
K
)
=
Φ
(
H
)
×
Φ
(
K
)


{\displaystyle \Phi (H\times K)=\Phi (H)\times \Phi (K)}

.
An example of a group with nontrivial Frattini subgroup is the cyclic group G of order




p

2




{\displaystyle p^{2}}

, where p is prime, generated by a, say; here,



Φ
(
G
)
=

⟨

a

p


⟩



{\displaystyle \Phi (G)=\left\langle a^{p}\right\rangle }

.




See also


Fitting subgroup
Frattini's argument
Socle




References



Hall, Marshall (1959). The Theory of Groups. New York: Macmillan. (See Chapter 10, especially Section 10.4.)

Kata Kunci Pencarian:

• Teorema Sylow
• Frattini subgroup
• Generating set of a group
• Maximal subgroup
• Frattini's argument
• Frattini
• Prüfer rank
• Rank of a group
• Sylow theorems
• Powerful p-group
• Inner automorphism