Frobenius group
Encyclopedia
In mathematics
, a Frobenius group is a transitive permutation group
on a finite set, such that no non-trivial element
fixes more than one point and some non-trivial element fixes a point.
They are named after F. G. Frobenius
.
H of a Frobenius group G fixing a point of the set X is called the Frobenius complement. The identity element together with all elements not in any conjugate of H form a normal subgroup
called the Frobenius kernel K. (This is a theorem due to Frobenius.) The Frobenius group G is the semidirect product
of K and H:
.
Both the Frobenius kernel and the Frobenius complement have very restricted structures. proved that the Frobenius kernel K is a nilpotent group
. If H has even order then K is abelian. The Frobenius complement H has the property that every subgroup whose order is the product of 2 primes is cyclic; this implies that its Sylow subgroups are cyclic
or generalized quaternion
groups. Any group such that all Sylow subgroups are cyclic is called a Z-group, and in particular must be a metacyclic group
: this means it is the extension of two cyclic groups. If a Frobenius complement H is not solvable then Zassenhaus
showed that it
has a normal subgroup of index
1 or 2 that is the product of SL2(5) and a metacyclic group of order coprime to 30. In particular, if a Frobenius complement coincides with its derived subgroup, then it is isomorphic with SL(2,5). If a Frobenius complement H is solvable then it has a normal metacyclic subgroup such that the quotient is a subgroup of the symmetric group on 4 points. A finite group is a Frobenius complement if and only if it has a faithful, finite-dimensional representation over a finite field in which non-identity group elements correspond to linear transformations without nonzero fixed points.
The Frobenius kernel K is uniquely determined by G as it is the Fitting subgroup
, and the Frobenius complement is uniquely determined up to conjugacy by the Schur-Zassenhaus theorem. In particular a finite group G is a Frobenius group in at most one way.
This definition is then generalized to the study of trivial intersection sets which allowed the results on Frobenius groups used in the classification of CA group
s to be extended to the results on CN group
s and finally the odd order theorem.
Assuming that
is the semidirect product
of the normal subgroup K and complement H, then the following restrictions on centralizers are equivalent to G being a Frobenius group with Frobenius complement H:
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a Frobenius group is a transitive permutation group
Permutation group
In mathematics, a permutation group is a group G whose elements are permutations of a given set M, and whose group operation is the composition of permutations in G ; the relationship is often written as...
on a finite set, such that no non-trivial element
fixes more than one point and some non-trivial element fixes a point.
They are named after F. G. Frobenius
Ferdinand Georg Frobenius
Ferdinand Georg Frobenius was a German mathematician, best known for his contributions to the theory of differential equations and to group theory...
.
Structure
The subgroupSubgroup
In group theory, given a group G under a binary operation *, a subset H of G is called a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...
H of a Frobenius group G fixing a point of the set X is called the Frobenius complement. The identity element together with all elements not in any conjugate of H form a normal subgroup
Normal subgroup
In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group. Normal subgroups can be used to construct quotient groups from a given group....
called the Frobenius kernel K. (This is a theorem due to Frobenius.) The Frobenius group G is the semidirect product
Semidirect product
In mathematics, specifically in the area of abstract algebra known as group theory, a semidirect product is a particular way in which a group can be put together from two subgroups, one of which is a normal subgroup. A semidirect product is a generalization of a direct product...
of K and H:
.Both the Frobenius kernel and the Frobenius complement have very restricted structures. proved that the Frobenius kernel K is a nilpotent group
Nilpotent group
In mathematics, more specifically in the field of group theory, a nilpotent group is a group that is "almost abelian". This idea is motivated by the fact that nilpotent groups are solvable, and for finite nilpotent groups, two elements having relatively prime orders must commute...
. If H has even order then K is abelian. The Frobenius complement H has the property that every subgroup whose order is the product of 2 primes is cyclic; this implies that its Sylow subgroups are cyclic
Cyclic group
In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g .-Definition:A group G is called cyclic if there exists an element g...
or generalized quaternion
Quaternion group
In group theory, the quaternion group is a non-abelian group of order eight, isomorphic to a certain eight-element subset of the quaternions under multiplication...
groups. Any group such that all Sylow subgroups are cyclic is called a Z-group, and in particular must be a metacyclic group
Metacyclic group
In group theory, a metacyclic group is an extension of a cyclic group by a cyclic group. That is, it is a group G for which there is a short exact sequence1 \rightarrow K \rightarrow G \rightarrow H \rightarrow 1,\,where H and K are cyclic...
: this means it is the extension of two cyclic groups. If a Frobenius complement H is not solvable then Zassenhaus
Hans Julius Zassenhaus
Hans Julius Zassenhaus was a German mathematician, known for work in many parts of abstract algebra, and as a pioneer of computer algebra....
showed that it
has a normal subgroup of index
Index of a subgroup
In mathematics, specifically group theory, the index of a subgroup H in a group G is the "relative size" of H in G: equivalently, the number of "copies" of H that fill up G. For example, if H has index 2 in G, then intuitively "half" of the elements of G lie in H...
1 or 2 that is the product of SL2(5) and a metacyclic group of order coprime to 30. In particular, if a Frobenius complement coincides with its derived subgroup, then it is isomorphic with SL(2,5). If a Frobenius complement H is solvable then it has a normal metacyclic subgroup such that the quotient is a subgroup of the symmetric group on 4 points. A finite group is a Frobenius complement if and only if it has a faithful, finite-dimensional representation over a finite field in which non-identity group elements correspond to linear transformations without nonzero fixed points.
The Frobenius kernel K is uniquely determined by G as it is the Fitting subgroup
Fitting subgroup
In mathematics, especially in the area of algebra known as group theory, the Fitting subgroup F of a finite group G, named after Hans Fitting, is the unique largest normal nilpotent subgroup of G. Intuitively, it represents the smallest subgroup which "controls" the structure of G when G is solvable...
, and the Frobenius complement is uniquely determined up to conjugacy by the Schur-Zassenhaus theorem. In particular a finite group G is a Frobenius group in at most one way.
Examples
- The smallest example is the symmetric group on 3 points, with 6 elements. The Frobenius kernel K has order 3, and the complement H has order 2.
- For every finite fieldFinite fieldIn abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...
Fq with q (> 2) elements, the group of invertible affine transformationAffine transformationIn geometry, an affine transformation or affine map or an affinity is a transformation which preserves straight lines. It is the most general class of transformations with this property...
s
, 
acting naturally on Fq is a Frobenius group. The preceding example corresponds to the case F3, the field with three elements.
- Another example is provided by the subgroup of order 21 of the collineation groupCollineationIn projective geometry, a collineation is a one-to-one and onto map from one projective space to another, or from a projective space to itself, such that the images of collinear points are themselves collinear. All projective linear transformations induce a collineation...
of the Fano planeFano planeIn finite geometry, the Fano plane is the finite projective plane with the smallest possible number of points and lines: 7 each.-Homogeneous coordinates:...
generated by a 3-fold symmetry σ fixing a point and a cyclic permutation τ of all 7 points, satisfying στ =τ²σ. Identifying F8* with the Fano plane, σ can be taken to be the restriction of the Frobenius automorphism σ(x)=x² of F8 and τ to be multiplication by any element not in the prime field F2 (i.e. a generator of the cyclic multiplicative group of F8). This Frobenius group acts simply transitively on the 21 flags in the Fano plane, i.e. lines with marked points.
- The dihedral groupDihedral groupIn mathematics, a dihedral group is the group of symmetries of a regular polygon, including both rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry.See also: Dihedral symmetry in three...
of order 2n with n odd is a Frobenius group with complement of order 2. More generally if K is any abelian group of odd order and H has order 2 and acts on K by inversion, then the semidirect productSemidirect productIn mathematics, specifically in the area of abstract algebra known as group theory, a semidirect product is a particular way in which a group can be put together from two subgroups, one of which is a normal subgroup. A semidirect product is a generalization of a direct product...
K.H is a Frobenius group.
- Many further examples can be generated by the following constructions. If we replace the Frobenius complement of a Frobenius group by a non-trivial subgroup we get another Frobenius group. If we have two Frobenius groups K1.H and K2.H then (K1 × K2).H is also a Frobenius group.
- If K is the non-abelian group of order 73 with exponent 7, and H is the cyclic group of order 3, then there is a Frobenius group G that is an extension K.H of H by K. This gives an example of a Frobenius group with non-abelian kernel. This was the first example of Frobenius group with nonabelian kernel (it was constructed by Otto Schmidt).
- If H is the group SL2(F5) of order 120, it acts fixed point freely on a 2-dimensional vector space K over the field with 11 elements. The extension K.H is the smallest example of a non-solvableSolvable groupIn mathematics, more specifically in the field of group theory, a solvable group is a group that can be constructed from abelian groups using extensions...
Frobenius group.
- The subgroup of a Zassenhaus groupZassenhaus groupIn mathematics, a Zassenhaus group, named after Hans Julius Zassenhaus, is a certain sort of doubly transitive permutation group very closely related to rank-1 groups of Lie type.- Definition :...
fixing a point is a Frobenius group.
- Frobenius groups whose Fitting subgroup has arbitrarily large nilpotency class were constructed by Ito: Let q be a prime power, d a positive integer, and p a prime divisor of q −1 with d ≤ p. Fix some field F of order q and some element z of this field of order p. The Frobenius complement H is the cyclic subgroup generated by the diagonal matrix whose i,ith entry is zi. The Frobenius kernel K is the Sylow q-subgroup of GL(d,q) consisting of upper triangular matrices with ones on the diagonal. The kernel K has nilpotency class d −1, and the semidirect product KH is a Frobenius group.
Representation theory
The irreducible complex representations of a Frobenius group G can be read off from those of H and K. There are two types of irreducible representations of G:- Any irreducible representation R of H gives an irreducible representation of G using the quotient map from G to H (that is, as a restricted representationRestricted representationIn mathematics, restriction is a fundamental construction in representation theory of groups. Restriction forms a representation of a subgroup from a representation of the whole group. Often the restricted representation is simpler to understand...
). These give the irreducible representations of G with K in their kernel. - If S is any non-trivial irreducible representation of K, then the corresponding induced representationInduced representationIn mathematics, and in particular group representation theory, the induced representation is one of the major general operations for passing from a representation of a subgroup H to a representation of the group G itself. It was initially defined as a construction by Frobenius, for linear...
of G is also irreducible. These give the irreducible representations of G with K not in their kernel.
Alternative definitions
There are a number of group theoretical properties which are interesting on their own right, but which happen to be equivalent to the group possessing a permutation representation that makes it a Frobenius group.- G is a Frobenius group if and only if G has a proper, nonidentity subgroup H such that H ∩ Hg is the identity subgroup for every g ∈ G − H.
This definition is then generalized to the study of trivial intersection sets which allowed the results on Frobenius groups used in the classification of CA group
CA group
In mathematics, in the realm of group theory, a group is said to be a CA-group or centralizer abelian group if the centralizer of any nonidentity element is an abelian subgroup. Finite CA-groups are of historical importance as an early example of the type of classifications that would be used in...
s to be extended to the results on CN group
CN group
In mathematics, in the area of algebra known as group theory, a more than fifty-year effort was made to answer a conjecture of : are all groups of odd order solvable? Progress was made by showing that CA-groups, groups in which the centralizer of a non-identity element is abelian, of odd order are...
s and finally the odd order theorem.
Assuming that
is the semidirect productSemidirect product
In mathematics, specifically in the area of abstract algebra known as group theory, a semidirect product is a particular way in which a group can be put together from two subgroups, one of which is a normal subgroup. A semidirect product is a generalization of a direct product...
of the normal subgroup K and complement H, then the following restrictions on centralizers are equivalent to G being a Frobenius group with Frobenius complement H:
- The centralizer CG(k) is a subgroup of K for every nonidentity k in K.
- CH(k) = 1 for every nonidentity k in K.
- CG(h) ≤ H for every nonidentity h in H.