Frattini subgroup

In mathematics, particularly in group theory, the Frattini subgroup

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

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] An example of a group with nontrivial Frattini subgroup is the cyclic group G of order

Hasse diagram of the lattice of subgroups of the dihedral group Dih 4 . In the second row are the maximal subgroups; their intersection (the Frattini subgroup ) is the central element in the third row. So Dih 4 has only one non-generating element beyond e .