View source for Frattini subgroup

[[File:Dih4 subgroups (cycle graphs).svg|thumb|360px|[[Hasse diagram]] of the [[lattice of subgroups]] of the [[dihedral group]] [[Dihedral group of order 8|Dih<sub>4</sub>.]] In the second row are the maximal subgroups; their intersection (the '''Frattini subgroup''') is the central element in the third row. So Dih<sub>4</sub> has only one non-generating element beyond ''e''.]]

In [[mathematics]], particularly in [[group theory]], the '''Frattini subgroup''' <math>\Phi(G)</math> of a [[group (mathematics)|group]] {{mvar|G}} is the [[intersection (set theory)|intersection]] of all [[maximal subgroup]]s 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 <math>\Phi(G)=G</math>. It is analogous to the [[Jacobson radical]] in the theory of [[ring (mathematics)|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.<ref>{{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}}</ref>

==Some facts==
* <math>\Phi(G)</math> 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 of a group|generating set]]; that is, an element ''a'' of {{mvar|G}} such that whenever {{mvar|X}} is a generating set of {{mvar|G}} containing ''a'', <math>X \setminus \{a\}</math> is also a generating set of {{mvar|G}}.
* <math>\Phi(G)</math> 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 <math>\Phi(G)</math> is [[nilpotent group|nilpotent]].
* If {{mvar|G}} is a finite [[p-group|''p''-group]], then <math>\Phi(G)=G^p [G,G]</math>. Thus the Frattini subgroup is the smallest (with respect to inclusion) [[normal subgroup]] ''N'' such that the [[quotient group]] <math>G/N</math> is an [[elementary abelian group]], i.e., [[Group isomorphism|isomorphic]] to a [[direct sum of abelian groups|direct sum]] of [[cyclic group]]s of [[Order (group theory)|order]] ''p''. Moreover, if the quotient group <math>G/\Phi(G)</math> (also called the ''Frattini quotient'' of {{mvar|G}}) has order <math>p^k</math>, 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]], <math>\Phi(G)=\{e\}</math>.
* If {{mvar|H}} and {{mvar|K}} are finite, then <math>\Phi(H\times K)=\Phi(H) \times \Phi(K)</math>.

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

==See also==
*[[Fitting subgroup]]
*[[Frattini's argument]]
*[[Socle (mathematics)|Socle]]

==References==
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* {{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.)
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[[Category:Group theory]]
[[Category:Functional subgroups]]