Frattini subgroup
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 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". 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 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 has order, then k is the smallest number of generators for . In particular, a finite p-group is cyclic if and only if its Frattini quotient is cyclic. A finite p-group is elementary abelian if and only if its Frattini subgroup is the trivial group,.
- If and are finite, then.