Frattini subgroup

In mathematics, particularly in group theory, the Frattini subgroup formula_1 of a group is the intersection of all maximal subgroups of . For the case that has no maximal subgroups, for example the trivial group {"e"} or the Prüfer group, it is defined by formula_2. 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 of order formula_13, where "p" is prime, generated by "a", say; here, formula_14.