In mathematics, particularly in group theory, the Frattini subgroup of a group is the intersection of all maximal subgroups of . For the case that has no maximal subgroups, for example the trivial group {e} or a Prüfer group, it is defined by . 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
- is equal to the set of all non-generators or non-generating elements of . A non-generating element of is an element that can always be removed from a generating set; that is, an element a of such that whenever is a generating set of containing a, is also a generating set of .
- is always a characteristic subgroup of ; in particular, it is always a normal subgroup of .
- If is finite, then is nilpotent.
- If is a finite p-group, then . Thus the Frattini subgroup is the smallest (with respect to inclusion) normal subgroup N such that the quotient group is an elementary abelian group, i.e., isomorphic to a direct sum of cyclic groups of order p. Moreover, if the quotient group (also called the Frattini quotient of ) has order , then k is the smallest number of generators for (that is, the smallest cardinality of a generating set for ). 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, .
- If and are finite, then .
An example of a group with nontrivial Frattini subgroup is the cyclic group of order , where p is prime, generated by a, say; here, .
See also
References
- (See Chapter 10, especially Section 10.4.)