In mathematics, particularly in group theory, the Frattini subgroup \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 \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

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,.