Frattini subgroup

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

In mathematics, particularly in group theory, the Frattini subgroup $\mathrm{\Phi }(G)$ 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 $\mathrm{\Phi }(G)=G$. 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

• $\mathrm{\Phi }(G)$ 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, $X\setminus \{a\}$ is also a generating set of G.
• $\mathrm{\Phi }(G)$ is always a characteristic subgroup of G; in particular, it is always a normal subgroup of G.
• If G is finite, then $\mathrm{\Phi }(G)$ is nilpotent.
• If G is a finite p-group, then $\mathrm{\Phi }(G)=G^p[G,G]$. Thus the Frattini subgroup is the smallest (with respect to inclusion) normal subgroup N such that the quotient group $G/N$ is an elementary abelian group, i.e., isomorphic to a direct sum of cyclic groups of order p. Moreover, if the quotient group $G/\mathrm{\Phi }(G)$ (also called the Frattini quotient of G) has order $p^k$, 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, $\mathrm{\Phi }(G)=\{e\}$.
• If H and K are finite, then $\mathrm{\Phi }(H\times K)=\mathrm{\Phi }(H)\times \mathrm{\Phi }(K)$.

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

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.)