University of Birmingham > Talks@bham > Birmingham and Warwick Algebra Seminar  > Enumerating transitive groups, bounding generator numbers, and complexity of algorithms in Computational Group Theory

Enumerating transitive groups, bounding generator numbers, and complexity of algorithms in Computational Group Theory

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If you have a question about this talk, please contact Gareth Tracey.

Computer databases that contain information on various types of groups can play a vital role in research. Examples include finite groups up to order 2000, primitive permutation groups up to degree 4095, and transitive permutation groups, recently extended to degree 48.

In the first part of the talk, we provide some details on the recent successful lengthy computer calculations involved in the enumeration of the 195,826,352 transitive groups of degree 48 (i.e. conjugacy classes in the symmetric group).

For a finitely generated group G, let d(G) be the smallest number of elements required to generate G.

In the second part of the talk, we survey results bounding d(G) for various types of finite permutation and matrix groups of a given degree, and describe how a knowledge of the transitive groups of degree 48 can be used to improve a result of Gareth Tracey bounding d(G) for transitive permutation groups.

Finally we describe recent results obtained jointly with Gareth, which bound d(G) log |G|, and are partly motivated by attempts to estimate the complexity of algorithms to compute the automorphism group of G.

This talk is part of the Birmingham and Warwick Algebra Seminar series.

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