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.
Statement Consider the group action of a permutation group G {\displaystyle G} acting on a set Ω {\displaystyle \Omega } . A base is a sequence of elements of Ω {\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 G {\displaystyle G} have the same size; The irredundant bases of G {\displaystyle G} are preserved by re-ordering; The irredundant bases of G {\displaystyle G} form the bases of a matroid.
References
Further reading https://www.theoremoftheday.org/GroupTheory/IBIS/TotDIBIS.pdf