A group is cyclic iff its subgroup lattice is distributive. Ore’s generalized one direction of this result. We will discuss a dual version of Ore’s result, for any boolean interval of finite groups under the assumption that the dual Euler totient of the interval is nonzero. We conjecture that the dual Euler totient is always nonzero for boolean intervals. We will discuss some techniques which may be helpful in proving it. We first see that dual Euler totient of an interval of finite groups is the Mobius invariant (upto a sign) of its coset poset P. Next in the boolean group complemented case, we prove that P is Cohen-Macaulay, using the existence of an explicit EL-labeling. We then see that nontrivial betti number of the order complex is nonzero, and so is the dual Euler totient.

- All seminars.
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Last updated: 02 Nov 2024