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Frattini subgroup

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Hasse diagram of the lattice of subgroups of the dihedral group Dih4. In the second row are the maximal subgroups; their intersection (the Frattini subgroup) is the central element in the third row. So Dih4 has only one non-generating element beyond e.

In mathematics, particularly in group theory, the Frattini subgroup Φ ( G ) {\displaystyle \Phi (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 a Prüfer group, it is defined by Φ ( G ) = G {\displaystyle \Phi (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 ) {\displaystyle \Phi (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 } {\displaystyle X\setminus \{a\}} is also a generating set of G.
  • Φ ( G ) {\displaystyle \Phi (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 ) {\displaystyle \Phi (G)} is nilpotent.
  • If G is a finite p-group, then Φ ( G ) = G p [ G , G ] {\displaystyle \Phi (G)=G^{p}} . Thus the Frattini subgroup is the smallest (with respect to inclusion) normal subgroup N such that the quotient group G / N {\displaystyle 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 ) {\displaystyle G/\Phi (G)} (also called the Frattini quotient of G) has order p k {\displaystyle 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 } {\displaystyle \Phi (G)=\{e\}} .
  • If H and K are finite, then Φ ( H × K ) = Φ ( H ) × Φ ( K ) {\displaystyle \Phi (H\times K)=\Phi (H)\times \Phi (K)} .

An example of a group with nontrivial Frattini subgroup is the cyclic group G of order p 2 {\displaystyle p^{2}} , where p is prime, generated by a, say; here, Φ ( G ) = ⟨ a p ⟩ {\displaystyle \Phi (G)=\left\langle a^{p}\right\rangle } .

See also

References

  1. Frattini, Giovanni (1885). "Intorno alla generazione dei gruppi di operazioni" (PDF). Accademia dei Lincei, Rendiconti. (4). I: 281–285, 455–457. JFM 17.0097.01.
  • Hall, Marshall (1959). The Theory of Groups. New York: Macmillan. (See Chapter 10, especially Section 10.4.)
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