Group of Frobenius

In Mathematical, a group of Frobenius is a Groupe of permutation Transitif on a Ensemble finished, such as any not-commonplace element does not fix more than one point and such as a certain element a point fixes. They were named in the honor of F.G. Frobenius.

Structure

The Sous-groupe H of a group of Frobenius G fixing a point of the unit X is called the complement of Frobenius . The neutral element at the same time as all the elements different from any combined H form a normal Sub-group called the core of Frobenius K . (This is a theorem due to Frobenius.) The group of Frobenius G is the semi-direct Produit of K and H :

G = K ⋊ H .

The core of Frobenius and the complement of Frobenius have both of the very restricted structures. J.G. Thompson showed that the core of Frobenius K is a Groupe nilpotent. If H is of an even nature then K is abelian. The complement of Frobenius H has the property which each sub-group whose order is the product of 2 prime numbers is cyclic; this implies that its sub-groups of Sylow is cyclic or of the groups of generalized quaternions. Any group such as all its sub-groups are cyclic is Métacyclique: this means that it is the extension of two cyclic groups. If a complement of Frobenius H is nonresolvable then Zassenhaus showed that it has a sub-group normal of index 1 or 2, i.e. the product of SL2 (5) and the group metacyclic of order first with 30. If a complement of Frobenius H is resolvable then it has a metacyclic sub-group normal such as the quotient is a sub-group of the symmetrical group at 4 points.

The core of Frobenius K is only determined by G as it is the Sous-groupe of Fitting and the complement of Frobenius only given until is combined by the Théorème of Schur-Zassenhaus. In particular, a group finished G is a group of Frobenius of to more the one manner.

Examples

the smallest example is the symmetrical group at 3 points, with 6 elements. The core of Frobenius K is of order 3 and the complement H is of a nature 2.

For each Body finished F_q with Q (> 2) elements, the group of the transformations closely connected invertible $X \ mapsto ax+b$ , $has \ 0$ with its natural action on F_q is a group of Frobenius. The preceding example corresponds to the case F₃ , the body with three elements.

more generally, the group of the triangular matrices higher 2 X 2 invertible of determinant 1 on any finished body of order at least 3 is a group of Frobenius. The core of Frobenius is the sub-group of the strictly higher triangular matrices (with the diagonal elements equal to 1), and the complement is the sub-group of the diagonal matrices.

the Groupe dihédral of order 2 N with odd N is a group of Frobenius with a complement of order 2. More generally, if K is an unspecified abelian group of an odd nature and H is of order 2 and acts on K by inversion, then the semi-direct Produit K.H is a group of Frobenius.

Beaucoup of advanced examples can be generated by following constructions. If we replace the complement of Frobenius of a group of Frobenius by a sub-group not-commonplace, we obtain another group of Frobenius. If we have two groups of Frobenius K ₁. H and K ₂. H then ( K ₁ × K ₂). H is also a group of Frobenius.

If K is the not-abelian group of order 7³ with exhibitor 7, and H is the cyclic group of order 3, then it exists a group of Frobenius G which is an extension K.H of H by K . This gives an example of a group of Frobenius with an not-abelian core.

If H is the group SSL ₂ ( F ₅) of order 120, it acts on the point freely fixed on a vector space at 2 dimensions K on the body at 11 elements. The extension K.H is the smallest example of a resolvable group of Frobenius not .

the sub-group of a Groupe of Zassenhaus fixing a point is a group of Frobenius.

Theory of the representation

The irreducible complex representations of a group of Frobenius G can be read distant from H and K . There exist two types of irreducible representations of G :

Any representation irreducible R of H gives an irreducible representation of G using the quotient application of G towards H (i.e., like a restricted Représentation). Those give the irreducible representations of G with K in their cores.
If S is an irreducible representation not-commonplace unspecified of K , then the induced Représentation corresponding of G is also irreducible. Those give the irreducible representations of G with K which are not in their cores.

References

D.S. Passman, Permutation groups , Benjamin 1968

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