Irredundant and greedy bases of primitive soluble groups
Algebra Seminar
21st January 2025, 4:00 pm – 5:00 pm
Fry Building, 2.04
Let G be a permutation group on a finite set X. An irredundant base for G is a sequence x_1, …, x_k of points from X such that each point x_i is moved by the pointwise stabiliser in G of x_1, …, x_(i-1), and the pointwise stabiliser in G of all k points is the identity. Bases are of critical importance in computational group theory, so upper bounds on both the longest and shortest sizes of an irredundant base are of interest. There is a natural greedy algorithm to construct a fairly small base, and in the late 1990s Peter Cameron conjectured that there is an absolute constant c such that if G is primitive then this algorithm constructs a base of length at most c times the minimum, whilst Akos Seress conjectured that if G is soluble then each greedy base has size at most four. We will present a survey of relevant results and then finish with a partial proof of one of these conjectures and a disproof of the other. The new results are joint with Sofia Brenner and Coen del Valle.
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