Frattini subgroup explained

In mathematics, particularly in group theory, the Frattini subgroup

\Phi(G) 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 \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.^[1]

Some facts

\Phi(G) 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, X \setminus \{a\} is also a generating set of .\Phi(G) is always a characteristic subgroup of ; in particular, it is always a normal subgroup of .

• If is finite, then

\Phi(G) is nilpotent.

• If 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) has order p ^{k} , 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, \Phi(G) = \{e\} .

• If and are finite, then

\Phi(H x K) = \Phi(H) x \Phi(K) .

An example of a group with nontrivial Frattini subgroup is the cyclic group of order

p ^{2} , where p is prime, generated by a, say; here, \Phi(G) = \left\langle a ^{p\right\rangle} .

See also

• Fitting subgroup
• Frattini's argument
• Socle

References

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

Notes and References

1. Giovanni. Frattini. Giovanni Frattini. Intorno alla generazione dei gruppi di operazioni. Accademia dei Lincei, Rendiconti . (4). I. 281–285, 455–457. 1885. 17.0097.01.