Assume that G is a finite non-abelian p-group. If G has an abelian maximal subgroup whose number of Generators is at least n, then G is called an M_n-group. For p = 2, M₂-groups have been classified. For odd prime p, this paper provides the isomorphism classification of M₂-groups, thereby achieving a complete classification of M₂-groups.

Assume that S is an nth-order complex sign pattern. If for every nth degree complex coefficient polynomial f(λ) with a leading coefficient of 1, there exists a complex matrix C\in Q\left(S\right) such that the characteristic polynomial of C is f(λ), then S is called a spectrally arbitrary complex sign pattern. That is, if the spectrum of nth-order complex sign pattern S is a set comprised of all spectra of nth-order complex matrices, then S is called a spectrally arbitrary complex sign pattern. This paper presents a class of spectrally arbitrary complex sign pattern with only 3n nonzero elements by adopting the method of Schur complement and row reduction.

Let F be a graph and H be a hypergraph. We say that H contains a Berge-F If there exists a bijection \phi: E(F)→E(H) such that for \mathrm{\forall }e\in E\left(F\right), e\subset \phi \left(e\right), and the Turán number of Berge-F is defined to be the maximum number of edges in an r-uniform hypergraph of order n that is Berge-F-free, denoted by ex_r(n, Berge-F). A linear forest is a graph whose connected components are all paths or isolated vertices. Let L_n,k be the family of all linear forests of n vertices with k edges. In this paper, Turán number of Berge-L_n,k in an r-uniform hypergraph is studied. When r⩾k +1 and 3 ⩽r⩽\lfloor \frac{k-1}{2}\rfloor -1, we determine the exact value of ex_r(n, Berge-L_n,k) respectively. When \lfloor \frac{k-1}{2}\rfloor⩽r⩽k, we determine the upper bound of ex_r(n, Berge-L_n,k).

This paper studies the properties of Nambu-Poisson geometry from the (n−1, k)-Dirac structure on a smooth manifold M. Firstly, we examine the automorphism group and infinitesimal on higher order Courant algebroid, to prove the integrability of infinitesimal Courant automorphism. Under the transversal smooth morphism \varphi :N\to M and anchor mapping of M on (n−1, k)-Dirac structure, it’s holds that the pullback (n−1, k)-Dirac structure on M turns out an (n−1, k)-Dirac structure on N. Then, given that the graph of Nambu-Poisson structure takes the form of (n−1, n−2)-Dirac structure, it follows that the single parameter variety of Nambu-Poisson structure is related to one variety closed n-symplectic form under gauge transformation. When \varphi :N\to Mis taken as the immersion mapping of (n−1)-cosymplectic submanifold, the pullback Nambu-Poisson structure on M turns out the Nambu-Poisson structure on N. Finally, we discuss the (n−1, 0)-Dirac structure on M can be integrated into a problem of (n−1)-presymplectic groupoid. Under the mapping \mathrm{\Pi }: M\to M/H, the corresponding (n−1, 0)-Dirac structure is F and E respectively. If E can be integrated into (n−1)-presymplectic groupoid (\mathfrak{g},\mathrm{\Omega }), then there exists the only \overline{\omega }, such that the corresponding integral of F is (n−1)-presymplectic groupoid \left(\overline{\mathfrak{g},}\overline{\omega }\right).