# Frattini subgroup

In mathematics, particularly in group theory, the **Frattini subgroup** [math]\displaystyle{ \Phi(G) }[/math] 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 [math]\displaystyle{ \Phi(G)=G }[/math]. 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.^{[1]}

## Some facts

- [math]\displaystyle{ \Phi(G) }[/math] 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*, [math]\displaystyle{ X \setminus \{a\} }[/math] is also a generating set of G. - [math]\displaystyle{ \Phi(G) }[/math] is always a characteristic subgroup of G; in particular, it is always a normal subgroup of G.
- If G is finite, then [math]\displaystyle{ \Phi(G) }[/math] is nilpotent.
- If G is a finite
*p*-group, then [math]\displaystyle{ \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]\displaystyle{ G/N }[/math] is an elementary abelian group, i.e., isomorphic to a direct sum of cyclic groups of order*p*. Moreover, if the quotient group [math]\displaystyle{ G/\Phi(G) }[/math] (also called the*Frattini quotient*of G) has order [math]\displaystyle{ p^k }[/math], 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, [math]\displaystyle{ \Phi(G)=\{e\} }[/math]. - If H and K are finite, then [math]\displaystyle{ \Phi(H\times K)=\Phi(H) \times \Phi(K) }[/math].

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

## See also

## References

- ↑ Frattini, Giovanni (1885). "Intorno alla generazione dei gruppi di operazioni".
*Accademia dei Lincei, Rendiconti*. (4)**I**: 281–285, 455–457. http://www.advgrouptheory.com/GTArchivum/Frattini/FrattiniPaper1885Transl.pdf.

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

Original source: https://en.wikipedia.org/wiki/Frattini subgroup.
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