Frobenius group
Suppose a finite group contains a subgroup satisfying specific properties. Using that information, what can be said about the structure of
itself? One way to tackle such a problem is via character theory (cf. also Character of a group), another is by viewing
as a permutation group. A classical and beautiful application of character theory is provided in elucidating the structure of Frobenius groups. Namely, let
. Assume that
whenever
. Then
is a so-called Frobenius complement in
; the group
is then a Frobenius group by definition. It was proved by G. Frobenius in 1901, see [a3], that the set
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is in fact a normal subgroup of . Almost a century later, Frobenius' proof that
is a subgroup of
is still the only existing proof; it uses character theory! The normal subgroup
is called the Frobenius kernel of
.
It can be shown that , that
and that the orders of
and
are relatively prime. Therefore, by the Schur–Zassenhaus theorem, all Frobenius complements in
are conjugate to each other. Below, let
be an element of a group and let
be a subset of that group; let
denote the set
.
A finite Frobenius group with Frobenius complement
and corresponding Frobenius kernel
satisfies:
1) for all
;
2) for all
;
3) for all
;
4) every is conjugate to an element of
;
5) if , then
is conjugate to every element of the coset
;
6) each non-principal complex irreducible character of induces irreducibly to
.
As a converse, assume that some finite group contains a normal subgroup
and some subgroup
satisfying
and
. Then the statements 1)–6) are all equivalent to each other, and if one of them is true, then
is a Frobenius complement of
, turning
into a Frobenius group with
as corresponding Frobenius kernel. Even more general, if some finite group
with proper normal subgroup
satisfies 1), then, applying one of the Sylow theorems, it is not hard to see that all orders of
and
are relatively prime. Whence there exists a subgroup
of
satisfying
and
(by the Schur–Zassenhaus theorem). Thus, again
is a Frobenius group with Frobenius complement
and Frobenius kernel
.
Viewed another way, suppose a finite group , containing a non-trivial proper subgroup
, acts transitively on a finite set
with
, such that
for some prescribed element
and such that only the identity of
leaves invariant more than one element of
. Then
is a Frobenius group with Frobenius complement
. Any element
of the Frobenius kernel
acts fixed-point freely on
, i.e.
for each
.
There is a characterization of finite Frobenius groups in terms of group characters only. Namely, let be a subgroup of a finite group
satisfying
. Then the following assertions are equivalent:
a) statement 6) above;
b) is a normal subgroup of
and
is a Frobenius group with Frobenius kernel
. The step from b) to a) was known to Frobenius; the converse step with, in addition,
normal in
is surely due to Frobenius; however, the step from a) to b) with
not necessarily normal in
is due to E.B. Kuisch (see [a7]).
This characterization led Kuisch, and later R.W. van der Waall, to the study of so-called -modular Frobenius groups; see [a8]. Namely, let
be a field of positive characteristic
. Then
is a
-modular Frobenius group if it contains a non-trivial normal subgroup
such that
is a splitting field for the group algebra
and if one of the following (equivalent) statements holds:
A) every non-principal irreducible -module
has the property that the induced
-module
is irreducible;
B) for every
-regular non-trivial element
. Any
featuring in A)–B) is a
-modular Frobenius kernel.
In 1959, J.G. Thompson [a9] showed that for a "classical" Frobenius group , the Frobenius kernel
is nilpotent (cf. also Nilpotent group), thereby solving a long-standing conjecture of W.S. Burnside. It was proved by H. Zassenhaus in 1939, [a10], that a Sylow
-subgroup (cf. also Sylow subgroup) of a Frobenius complement
of
is cyclic (cf. also Cyclic group) when
is odd, and cyclic or generalized quaternion if
. He also proved that if
is not solvable (cf. also Solvable group), then it admits precisely one non-Abelian composition factor, namely the alternating group on five symbols.
The situation is more involved for -modular Frobenius groups. Namely, a
-modular Frobenius kernel
is either solvable (cf. also Solvable group) or else
and any non–Abelian composition factor of
is isomorphic to
for some integer
.
Furthermore, assume that is not a
-group. Then:
any Sylow -subgroup of
is cyclic whenever
is relatively prime to
;
any Sylow -subgroup of
is cyclic or generalized quaternion if
is odd. On the other hand, any non-trivial finite
-group (
a prime number) is isomorphic to some quotient group
, where
is a suitable
-modular Frobenius group with
-modular Frobenius kernel
. See also [a8].
Historically, finite Frobenius groups have played a major role in many areas of group theory, notably in the analysis of -transitive groups and finite simple groups (cf. also Transitive group; Simple finite group).
Frobenius groups can be defined for infinite groups as well. Those groups are the non-regular transitive permutation groups in which only the identity has more than one fixed point. Again, let consist of the identity and those elements of the Frobenius group
not occurring in any point stabilizer (cf. also Stabilizer). Contrary to the finite case, it is now not always true that
is a subgroup of
. See [a2] for examples.
References
[a1] | Yu.G. Berkovich, E.M. Zhmud, "Characters of finite groups" , Amer. Math. Soc. (1998/9) |
[a2] | J.D. Dixon, B. Mortimer, "Permutation groups" , GTM , 163 , Springer (1996) |
[a3] | G. Frobenius, "Ueber auflösbare Gruppen IV" Sitzungsber. Preuss. Akad. Wissenschaft. (1901) pp. 1216–1230 |
[a4] | B. Huppert, "Endliche Gruppen" , I , Springer (1967) |
[a5] | B. Huppert, "Character theory of finite groups" , Experim. Math. , 25 , de Gruyter (1998) |
[a6] | I.M. Isaacs, "Character theory of finite groups" , Acad. Press (1976) |
[a7] | E.B. Kuisch, R.W. van der Waall, "Homogeneous character induction" J. Algebra , 156 (1993) pp. 395–406 |
[a8] | E.B. Kuisch, R.W. van der Waall, "Modular Frobenius groups" Manuscripta Math. , 90 (1996) pp. 403–427 |
[a9] | J.G. Thompson, "Finite groups with fixed point free automorphisms of prime order" Proc. Nat. Acad. Sci. USA , 45 (1959) pp. 578–581 |
[a10] | H. Zassenhaus, "Ueber endliche Fastkörper" Abh. Math. Sem. Univ. Hamburg , 11 (1936) pp. 187–220 |
Frobenius group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Frobenius_group&oldid=19132