Frattini subgroup

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 has no maximal subgroups, for example the trivial group e or the Prüfer group, it is defined by Φ(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.

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, ${\displaystyle \Phi (G)=\left\langle a^{p}\right\rangle }$.

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