Frattini subgroup



In mathematics, particularly in group theory, the Frattini subgroup $$\Phi(G)$$ 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 $$\Phi(G)=G$$. 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

 * $$\Phi(G)$$ 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, $$X \setminus \{a\}$$ is also a generating set of $G$.
 * $$\Phi(G)$$ is always a characteristic subgroup of $G$; in particular, it is always a normal subgroup of $G$.
 * If $G$ is finite, then $$\Phi(G)$$ is nilpotent.
 * If $G$ 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 $G$) 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, $$\Phi(G)=\{e\}$$.
 * If $H$ and $K$ 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 $G$ of order $$p^2$$, where p is prime, generated by a, say; here, $$\Phi(G)=\left\langle a^p\right\rangle$$.