Sharply 3-transitive Groups with Finite Element

In this paper we study sharply 3-transitive groups. The local finiteness of sharply triply transitive permutation groups of characteristic p > 3 containing a finite element of order p is proved. Keywords: group, sharply k-transitive group, sharply 3-transitive group, locally finite group, neardomain, near-field

The local structure of the free boundary in the fractional obstacle problem

Building upon the recent results in [M. Focardi and E. Spadaro, On the measure and the structure of the free boundary of the lower-dimensional obstacle problem, Arch. Ration. Mech. Anal. 230 2018, 1, 125–184] we provide a thorough description of the free boundary for the solutions to the fractional obstacle problem in {\mathbb{R}^{n+1}} with obstacle function φ (suitably smooth and decaying fast at infinity) up to sets of null {{\mathcal{H}}^{n-1}} measure. In particular, if φ is analytic, the problem reduces to the zero obstacle case dealt with in [M. Focardi and E. Spadaro, On the measure and the structure of the free boundary of the lower-dimensional obstacle problem, Arch. Ration. Mech. Anal. 230 2018, 1, 125–184] and therefore we retrieve the same results:(i)local finiteness of the {(n-1)}-dimensional Minkowski content of the free boundary (and thus of its Hausdorff measure),(ii){{\mathcal{H}}^{n-1}}-rectifiability of the free boundary,(iii)classification of the frequencies and of the blowups up to a set of Hausdorff dimension at most {(n-2)} in the free boundary.Instead, if {\varphi\in C^{k+1}(\mathbb{R}^{n})}, {k\geq 2}, similar results hold only for distinguished subsets of points in the free boundary where the order of contact of the solution with the obstacle function φ is less than {k+1}.

Matric generators of coalgebras and bialgebras

Takeuchi asserted that if a bialgebra [Formula: see text] over a field [Formula: see text] is finitely generated as a [Formula: see text]-algebra, then [Formula: see text] is a matric bialgebra. We introduce the notion of a matric coalgebra over a commutative ring [Formula: see text]. We show that if [Formula: see text] is faithfully projective as a [Formula: see text]-module, then [Formula: see text] is a matric coalgebra. Using this, we also show that if a bialgebra [Formula: see text] over a semihereditary ring [Formula: see text] is projective as a [Formula: see text]-module, then any finite subset of [Formula: see text] is contained in some matric subbialgebra. This result is a generalization of Takeuchi’s assertion and can be regarded as a local finiteness theorem on bialgebras.