Cameron–Fon-Der-Flaass IBIS theorem

In mathematics, the Cameron–Fon-Der-Flaass IBIS theorem bridges algebraic combinatorics and group theory. The theorem was discovered in 1995 by two mathematicians Peter Cameron and Dima Von-Der-Flaass.^[1][2]

Consider the group action of a permutation group ${\displaystyle G}$ acting on a set ${\displaystyle \Omega }$. A base is a sequence of elements of ${\displaystyle \Omega }$ which, when fixed, destroys all symmetry, i.e. its pointwise stabilizer is trivial. A base is irredundant if each element further reduces symmetry, i.e. no element in the sequence is fixed by the pointwise stabiliser of its predecessors. Now the following are equivalent:

• All irredundant bases of ${\displaystyle G}$ have the same size;
• The irredundant bases of ${\displaystyle G}$ are preserved by re-ordering;
• The irredundant bases of ${\displaystyle G}$ form the bases of a matroid.

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

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