Frattini subgroup

Hasse diagram of the lattice of subgroups of the dihedral group Dih₄

In 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.

Some facts

Φ( $G$ ) is equal to the set of all non-generators or non-generating elements of $G$ . A non-generating element of $G$ is an element that can always be removed from a generating set; that is, an element a of $G$ such that whenever $X$ is a generating set of $G$ containing a, $X$ − {a} is also a generating set of $G$ .
Φ( $G$ ) is always a characteristic subgroup of $G$ ; in particular, it is always a normal subgroup of $G$ .
If $G$ is finite, then Φ( $G$ ) is nilpotent.
If $G$ is a finite p-group, then Φ( $G$ ) = $G$ ^p [ $G$ , $G$ ]. Thus the Frattini subgroup is the smallest (with respect to inclusion) normal subgroup N such that the quotient group $G$ /N is an elementary abelian group, i.e., isomorphic to a direct sum of cyclic groups of order p. Moreover, if the quotient group $G$ /Φ( $G$ ) (also called the Frattini quotient of $G$ ) has order p^k, then k is the smallest number of generators for $G$ (that is the smallest cardinality of a generating set for $G$ ). In particular a finite p-group is cyclic if and only if its Frattini quotient is cyclic (of order p). A finite p-group is elementary abelian if and only if its Frattini subgroup is the trivial group, Φ( $G$ ) = e.
If $H$ and $K$ are finite, then $Φ(H \times K) = Φ(H) \times Φ(K)$ .

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

References

Hall, Marshall (1959). The Theory of Groups. New York: Macmillan. (See Chapter 10, especially Section 10.4.)

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Frattini subgroup

Some facts

See also

References