HOME

TheInfoList



Click Here for Items Related To - Frattini subgroup
OR:
Frattini subgroup on:   [Wikipedia]   [Google]   [Amazon]

In mathematics, particularly in
group theory In abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as ...
, the Frattini subgroup \Phi(G) of a
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
is the intersection of all maximal subgroups of . For the case that has no maximal subgroups, for example the
trivial group In mathematics, a trivial group or zero group is a group consisting of a single element. All such groups are isomorphic, so one often speaks of the trivial group. The single element of the trivial group is the identity element and so it is usuall ...
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 . 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 \ is also a generating set of . * \Phi(G) is always a
characteristic subgroup In mathematics, particularly in the area of abstract algebra known as group theory, a characteristic subgroup is a subgroup that is mapped to itself by every automorphism of the parent group. Because every conjugation map is an inner automorphis ...
of ; in particular, it is always a normal subgroup of . * If is finite, then \Phi(G) is
nilpotent In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0. The term was introduced by Benjamin Peirce in the context of his work on the cla ...
. * If is a finite ''p''-group, then \Phi(G)=G^p ,G/math>. 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 In mathematics, specifically in group theory, an elementary abelian group (or elementary abelian ''p''-group) is an abelian group in which every nontrivial element has order ''p''. The number ''p'' must be prime, and the elementary abelian grou ...
, i.e., isomorphic to a
direct sum The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a more ...
of cyclic groups of
order Order, ORDER or Orders may refer to: * Categorization, the process in which ideas and objects are recognized, differentiated, and understood * Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of ...
''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 In mathematics, a trivial group or zero group is a group consisting of a single element. All such groups are isomorphic, so one often speaks of the trivial group. The single element of the trivial group is the identity element and so it is usuall ...
, \Phi(G)=\. * If and 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 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

* (See Chapter 10, especially Section 10.4.) {{DEFAULTSORT:Frattini Subgroup Group theory Functional subgroups