Frattini subgroup

In mathematics, particularly in group theory, the Frattini subgroup ${\displaystyle \Phi (G)}$ 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 the Prüfer group, it is defined by ${\displaystyle \Phi (G)=G}$. 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.[1]

Hasse diagram of the lattice of subgroups of the dihedral group Dih₄. In the 3-element layer are the maximal subgroups; their intersection (the Frattini subgroup) is the central element in the 5-element layer. So Dih₄ has only one non-generating element beyond e.

Some facts

• ${\displaystyle \Phi (G)}$ 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, ${\displaystyle X\setminus \{a\}}$ is also a generating set of 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 ${\displaystyle \Phi (G)}$ is nilpotent.

• If G is a finite p-group, then ${\displaystyle \Phi (G)=G^{p}[G,G]}$. Thus the Frattini subgroup is the smallest (with respect to inclusion) normal subgroup N such that the quotient group ${\displaystyle G/N}$ is an elementary abelian group, i.e., isomorphic to a direct sum of cyclic groups of order p. Moreover, if the quotient group ${\displaystyle G/\Phi (G)}$ (also called the Frattini quotient of G) has order ${\displaystyle p^{k}}$, 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, ${\displaystyle \Phi (G)=\{e\}}$.

• If H and K are finite, then ${\displaystyle \Phi (H\times K)=\Phi (H)\times \Phi (K)}$.

An example of a group with nontrivial Frattini subgroup is the cyclic group G of order ${\displaystyle p^{2}}$, where p is prime, generated by a, say; here, ${\displaystyle \Phi (G)=\left\langle a^{p}\right\rangle }$.

See also

• Fitting subgroup
• Frattini's argument
• Socle

References

1. Frattini, Giovanni (1885). "Intorno alla generazione dei gruppi di operazioni" (PDF). Accademia dei Lincei, Rendiconti. (4). I: 281–285, 455–457. JFM 17.0097.01.

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