Frattini subgroup

Hasse diagram of the lattice of subgroups of the dihedral group Dih4In 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 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.

单词	Frattini subgroup
释义	Frattini subgroup 英语百科 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.
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