Frattini subgroup

Hasse diagram of the lattice of subgroups of the dihedral group Dih₄

In the 3-element layer are the maximal subgroups;
their intersection (the F. s.) is the central element in the 5-element layer.
So Dih₄ has only one non-generating element beyond e.

In mathematics, the Frattini subgroup Φ(G) of a group G is the intersection of all maximal subgroups of G. For the case that G is the trivial group e, which has no maximal subgroups, it is defined by Φ(e) = e. 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) 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, X − {a} is also a generating set of G.
• Φ(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) is nilpotent.
• If G is a finite p-group, then Φ(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/Φ(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, Φ(G) = e.

An example of a group with nontrivial Frattini subgroup is the cyclic group G of order p², where p is prime, generated by a, say; here, $\Phi(G)=\left\langle a^p\right\rangle$.

See also

• Fitting subgroup
• Socle

Reference

• Hall, Marshall (1959). The theory of groups. New York, N.Y.: Macmillan.  (See Chapter 10, especially Section 10.4.)