Cameron–Fon-Der-Flaass IBIS theorem explained

In mathematics, Cameron–Fon-Der-Flaass IBIS theorem arises in the dynamical algebraic combinatorics. The theorem was discovered in 1995 by two mathematicians Peter Cameron and Dima Fon-Der-Flaas.^[1] The theorem is considered to be a link between group theory and graph theory as it studies redundancy of a group.^[2]

Statement

Let

P be a permutational group of \Omega , then the following are equivalent:

• Irredundant bases of

P are stored by re-ordering.

• The bases of matroid are formed due to the irredundant bases of

P .

• Every irredundant base of

P got the same size.

References

1. Patrias . Rebecca . Pechenik . Oliver . 2020 . Dynamics of plane partitions: Proof of the Cameron–Fon-Der-Flaass conjecture . Forum of Mathematics, Sigma . en . 8 . 62 . 10.1017/fms.2020.61 . 2050-5094. 2003.13152 .
2. Cameron . P. J . Fon-Der-Flaass . D. G . 1995-11-01 . Bases for permutation groups and matroids . European Journal of Combinatorics . en . 16 . 6 . 537–544 . 10.1016/0195-6698(95)90035-7 . 0195-6698.

Further reading

• https://www.theoremoftheday.org/GroupTheory/IBIS/TotDIBIS.pdf