Solvable flag transitive affine groups

David A. Foulser

doi:10.1007/bf01110390

Abelian simply transitive affine groups of symplectic type

Annales de l’institut Fourier ◽

10.5802/aif.1932 ◽

2002 ◽

Vol 52 (6) ◽

pp. 1729-1751 ◽

Cited By ~ 2

Author(s):

Oliver Baues ◽

Vicente Cortés

Keyword(s):

Affine Groups ◽

Transitive Affine

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Permutation representations and rational irreducibility

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700038508 ◽

2005 ◽

Vol 71 (3) ◽

pp. 493-503 ◽

Cited By ~ 5

Author(s):

John D. Dixon

Keyword(s):

Permutation Group ◽

Partial Information ◽

Transitive Group ◽

Transitive Permutation Group ◽

Rational Representation ◽

Affine Groups ◽

Transitive Affine ◽

Affine Type

The natural character π of a finite transitive permutation group G has the form 1G + θ where θ is a character which affords a rational representation of G. We call G a QI-group if this representation is irreducible over ℚ. Every 2-transitive group is a QI-group, but the latter class of groups is larger. It is shown that every QI-group is ¾-transitive and primitive, and that it is either almost simple or of affine type. QI-groups of affine type are completely determined relative to the 2-transitive affine groups, and partial information is obtained about the socles of simply transitive almost simple QI-groups. The only known simply transitive almost simple QI-groups are of degree 2k−1(2k − 1) with 2k − 1 prime and socle isomorphic to PSL(2, 2k).

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Affine Sets and Affine Groups

10.1017/cbo9781107325456 ◽

1980 ◽

Cited By ~ 7

Author(s):

D. G. Northcott

Keyword(s):

Affine Groups

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Two Interacting Coordinate Hopf Algebras of Affine Groups of Formal Series on a Category

Algebra ◽

10.1155/2013/370618 ◽

2013 ◽

Vol 2013 ◽

pp. 1-10

Author(s):

Laurent Poinsot

Keyword(s):

Power Series ◽

Formal Power Series ◽

Semidirect Product ◽

Hopf Algebras ◽

Formal Series ◽

Product Structure ◽

Locally Finite ◽

Commutative Algebras ◽

Affine Groups ◽

Formal Power

A locally finite category is defined as a category in which every arrow admits only finitely many different ways to be factorized by composable arrows. The large algebra of such categories over some fields may be defined, and with it a group of invertible series (under multiplication). For certain particular locally finite categories, a substitution operation, generalizing the usual substitution of formal power series, may be defined, and with it a group of reversible series (invertible under substitution). Moreover, both groups are actually affine groups. In this contribution, we introduce their coordinate Hopf algebras which are both free as commutative algebras. The semidirect product structure obtained from the action of reversible series on invertible series by anti-automorphisms gives rise to an interaction at the level of their coordinate Hopf algebras under the form of a smash coproduct.

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