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Transitive Groups Naming Scheme

Description

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This page briefly describes some of the notations introduced in the paper "On Transitive Permutation Groups" by J.H. Conway, A. Hulpke, and J. McKay, LMS J. Comput. Math. 1 (1998), 1-8. These notations are reminiscent of the notations used in "The Atlas of Finite Groups" by J.H. Conway, R.T. Curtis, S.P. Norton, R.A. Parker, and R.A. Wilson.

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These notations are used by the galois function.

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Capital letters denote families of groups:

A   : Alternating

F   : Frobenius

E   : Elementary

S   : Symmetric

AL  : Affine linear

C   : Cyclic

M   : Mathieu

D   : Dihedral

Q_8 : Quaternionic group

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Except for dihedral and Frobenius group, a name of the form Xn, where X is a family name, denotes the n-th member of this family acting as a permutation group on n points. For instance, S3 is the symmetric group on 3 elements. Moreover, Xn denotes the same abstract group, but not necessarily with the same action. For instance, A46 is the alternating group on 4 elements acting transitively on a set of 6 elements. For dihedral and Frobenius groups, Fn or Dn denotes the group of order n. For instance, D4=D84 and D66 is the dihedral group with six elements acting transitively on a set of 6 elements.

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An integer n stands for a cyclic group with n elements.

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Let X and Y be groups. Then

  

XY or X.Y indicates a group with a normal subgroup of structure X, for which the corresponding quotient has structure Y.

  

X specifies that the  group is a split extension.

  

 XxY denotes a direct product where the action is the natural action on the Cartesian product of the sets.

  

 X[1m]Y denotes a subdirect product corresponding to two epimorphisms e1: XF and e2: YF where F is a group of order m. In other words, the group consists of elements a,b in the direct product XxY such that e1a=e2b.

  

Xn is the direct product of n groups of structure X.

  

 XwrY denotes a wreath product.

  

 [X]Y is an imprimitive group derived from a semi-direct product. The group X is the intersection of the block stabilizers. See the paper by Conway, Hulpke, and McKay for more information. In particular [Xn]Y (where Y has degree n) is the permutational wreath product XwrY.

  

 1mXY denotes a subgroup of [X]Y. There exists two epimorphisms e1: XF and e2: YF (where the order of F is m, such that the group consists of elements a,b in [X]Y satisfying e1a=e2b.

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Lower case letters are used to distinguish different groups arising from the same general construction. See the paper by Conway, Hulpke, and McKay for more information.

See Also

combinat, group, group[transgroup], http://web.mat.bham.ac.uk/atlas/v2.0/


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