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. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups. In Mathematics, a permutation group is a group G whose elements are Permutations of a given set M, and whose group operation In Mathematics, a set is called finite if there is a Bijection between the set and some set of the form {1 2. They are named after F. G. Frobenius. Ferdinand Georg Frobenius ( October 26, 1849 – August 3, 1917) was a German Mathematician, best-known for his contributions

Contents 1 Structure 2 Examples 3 Representation theory 4 Alternative definitions 5 References

Structure

The subgroup H of a Frobenius group G fixing a point of the set X is called the Frobenius complement. In Group theory, given a group G under a Binary operation * we say that some Subset H of G is a subgroup of The identity element together with all elements not in any conjugate of H form a normal subgroup called the Frobenius kernel K. In Mathematics, more specifically in Abstract algebra, a normal subgroup is a special kind of Subgroup. (This is a theorem due to Frobenius. ) The Frobenius group G is the semidirect product of K and H:

G = K ⋊ H. In Mathematics, especially in the area of Abstract algebra known as Group theory, a semidirect product is a particular way in which a group can

Both the Frobenius kernel and the Frobenius complement have very restricted structures. J. G. Thompson proved that the Frobenius kernel K is a nilpotent group. John Griggs Thompson (born October 13 1932 in Ottawa Kansas, USA) is a Mathematician noted for his work in the field of Finite In Group theory, a nilpotent group is a group having a special property that makes it "almost" abelian, through repeated application of the 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. In Mathematics, specifically Group theory, the Sylow theorems, named after Ludwig Sylow, form a partial converse to Lagrange's theorem, which In Group theory, a cyclic group or monogenous group is a group that can be generated by a single element in the sense that the group has an In Group theory, the quaternion group is a non-abelian group of order 8 Any group such that all Sylow subgroups are cyclic is a metacyclic group: this means it is the extension of two cyclic groups. In Group theory, a metacyclic group is an extension of a Cyclic group by a cyclic group 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. Hans Julius Zassenhaus ( 28 May 1912 &ndash 21 November 1991) was a German Mathematician, known for work in many parts of 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.

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 Mathematics, especially in the area of algebra known as Group theory, the Fitting subgroup F of a finite group The Schur–Zassenhaus theorem is a Theorem in Group theory which states that if G is a finite group, and N is a Normal subgroup 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 field F_q with q (> 2) elements, the group of invertible affine transformations , acting naturally on F_q is a Frobenius group. In Abstract algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains only finitely many elements In Geometry, an affine transformation or affine map or an affinity (from the Latin affinis, "connected with" between two Vector 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 group of the Fano plane generated by a 3-fold symmetry σ fixing a point and a cyclic permutation τ of all 7 points, satisfying στ =τ²σ. A collineation is a one-to-one map from one Projective space to another or from a Projective plane onto itself such that the images of collinear points are themselves In Finite geometry, the Fano plane (after Gino Fano) is the Projective plane with the least number of points and lines 7 each 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. In Commutative algebra and field theory, which are branches of Mathematics, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a In Mathematics, the characteristic of a ring R, often denoted char( R) is defined to be the smallest number of times one must add the ring's e. a generator of the cyclic multiplicative group of F₈). In Abstract algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains only finitely many elements This Frobenius group acts simply transitively on the 21 flags in the Fano plane, i. In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups. e. lines with marked points.

• The dihedral group of order 2n with n odd is a Frobenius group with complement of order 2. In Mathematics, a dihedral group is the group of symmetries of a Regular polygon, including both rotations and reflections 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 product K. In Mathematics, especially in the area of Abstract algebra known as Group theory, a semidirect product is a particular way in which a group can 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.

• 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-solvable Frobenius group. In the history of Mathematics, the origins of Group theory lie in the search for a proof of the general unsolvability of Quintic and higher equations finally

• The subgroup of a Zassenhaus group fixing a point is a Frobenius group. In Mathematics, a Zassenhaus group, named after Hans Julius Zassenhaus, is a certain sort of doubly transitive Permutation group very closely

• 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,i'th 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. In Mathematics, Representation theory is a technique for analyzing abstract groups in terms of groups of Linear transformations See the article on 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 representation). In the mathematical field of Representation theory, group representations describe abstract groups in terms of Linear transformations of In Mathematics, restriction is a fundamental construction in representation theory of groups Restriction forms a representation of a subgroup from a representation 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 representation of G is also irreducible. In Mathematics, and in particular Group representation theory the induced representation is one of the major general operations for passing from a representation 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 ∩ H^g 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 groups to be extended to the results on CN groups and finally the odd order theorem. 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 The CN Group Limited is an independent local media business based in Carlisle which operates in three different media fields In Mathematics, the Feit–Thompson theorem, or odd order theorem, states that every finite group of odd order is solvable.

Assuming that G = K ⋊ H 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:

• The centralizer C_G(k) is a subgroup of K for every nonidentity k in K
• C_H(k) = 1 for every nonidentity k in K
• C_H(h) ≤ H for every nonidentity h in H

References

• D. In Mathematics, especially in the area of Abstract algebra known as Group theory, a semidirect product is a particular way in which a group can In Group theory, the centralizer and normalizer of a Subset S of a group G are Subgroups of G which In Group theory, the centralizer and normalizer of a Subset S of a group G are Subgroups of G which S. Passman, Permutation groups, Benjamin 1968
• I. M. Isaacs, Character theory of finite groups, AMS Chelsea 1976
• B. Huppert, Endliche Gruppen I, Springer 1967

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