Frattini subgroup

File:Dih4 subgroups (cycle graphs).svg 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 $\Phi(G)$ of a group {{mvar|G}} is the intersection of all maximal subgroups of {{mvar|G}}. For the case that {{mvar|G}} has no maximal subgroups, for example the trivial group {e} or a Prüfer group, it is defined by $\Phi(G)=G$ . 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.{{cite journal|first=Giovanni|last= Frattini|author-link=Giovanni Frattini|title=Intorno alla generazione dei gruppi di operazioni|journal=Accademia dei Lincei, Rendiconti |series=(4)|volume= I|pages=281–285, 455–457|year= 1885| jfm=17.0097.01| url=http://www.advgrouptheory.com/GTArchivum/Frattini/FrattiniPaper1885Transl.pdf}}

Some facts

$\Phi(G)$ is equal to the set of all non-generators or non-generating elements of {{mvar|G}}. A non-generating element of {{mvar|G}} is an element that can always be removed from a generating set; that is, an element a of {{mvar|G}} such that whenever {{mvar|X}} is a generating set of {{mvar|G}} containing a, $X \setminus \{a\}$ is also a generating set of {{mvar|G}}.
$\Phi(G)$ is always a characteristic subgroup of {{mvar|G}}; in particular, it is always a normal subgroup of {{mvar|G}}.
If {{mvar|G}} is finite, then $\Phi(G)$ is nilpotent.
If {{mvar|G}} is a finite p-group, then $\Phi(G)=G^p [G,G]$ . 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 $G/\Phi(G)$ (also called the Frattini quotient of {{mvar|G}}) has order $p^k$ , then k is the smallest number of generators for {{mvar|G}} (that is, the smallest cardinality of a generating set for {{mvar|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, $\Phi(G)=\{e\}$ .
If {{mvar|H}} and {{mvar|K}} are finite, then $\Phi(H\times K)=\Phi(H) \times \Phi(K)$ .

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

References

{{Cite book | last = Hall | first = Marshall | author-link = Marshall Hall (mathematician) | title = The Theory of Groups | publisher = Macmillan | date = 1959 | location = New York }} (See Chapter 10, especially Section 10.4.)

Category:Group theory

Category:Functional subgroups

Frattini subgroup

Some facts

See also

References