HOME

TheInfoList



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

In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, particularly in
group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ( ...
, the Frattini subgroup \Phi(G) of a group is the
intersection In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their ...
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.


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 of ; in particular, it is always a
normal subgroup In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group ...
of . * If is finite, then \Phi(G) is
nilpotent In mathematics, an element x of a ring (mathematics), 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, along with its sister Idempotent (ring theory), idem ...
. * 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 In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group ...
''N'' such that the
quotient group A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored out"). For ex ...
G/N is an
elementary abelian group In mathematics, specifically in group theory, an elementary abelian group is an abelian group in which all elements other than the identity have the same order. This common order must be a prime number, and the elementary abelian groups in whic ...
, i.e.,
isomorphic In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
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. As an example, the direct sum of two abelian groups A and B is anothe ...
of
cyclic group In abstract algebra, a cyclic group or monogenous group is a Group (mathematics), group, denoted C_n (also frequently \Z_n or Z_n, not to be confused with the commutative ring of P-adic number, -adic numbers), that is Generating set of a group, ge ...
s 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 In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
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)=\. * 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 In abstract algebra, a cyclic group or monogenous group is a Group (mathematics), group, denoted C_n (also frequently \Z_n or Z_n, not to be confused with the commutative ring of P-adic number, -adic numbers), that is Generating set of a group, ge ...
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