A bound for the class of nilpotent symplectic alternating algebras

We continue developing the theory of nilpotent symplectic alternating algebras. The algebras of upper bound nilpotent class, that we call maximal algebras, have been introduced and well studied. In this paper, we continue with the external case problem of algebras of minimal nilpotent class. We show the existence of a subclass of algebras over a field [Formula: see text] that is of certain lower bound class that depends on the dimension only. This suggests a minimal bound for the class of nilpotent algebras of dimension [Formula: see text] of rank [Formula: see text] over any field.

FINITE -GROUPS ALL OF WHOSE NONNORMAL SUBGROUPS HAVE BOUNDED NORMAL CORES

Given a positive integer $m$, a finite $p$-group $G$ is called a $BC(p^{m})$-group if $|H_{G}|\leq p^{m}$ for every nonnormal subgroup $H$ of $G$, where $H_{G}$ is the normal core of $H$ in $G$. We show that $m+2$ is an upper bound for the nilpotent class of a finite $BC(p^{m})$-group and obtain a necessary and sufficient condition for a $p$-group to be of maximal class. We also classify the $BC(p)$-groups.

Groups of automorphisms of local fields of period p and nilpotent class < p, II

Suppose [Formula: see text] is a finite field extension of [Formula: see text] containing a primitive [Formula: see text]th root of unity. Let [Formula: see text] be the maximal quotient of period [Formula: see text] and nilpotent class [Formula: see text] of the Galois group of a maximal [Formula: see text]-extension of [Formula: see text]. We describe the ramification filtration [Formula: see text] and relate it to an explicit form of the Demushkin relation for [Formula: see text]. The results are given in terms of Lie algebras attached to the appropriate [Formula: see text]-groups by the classical equivalence of the categories of [Formula: see text]-groups and Lie algebras of nilpotent class [Formula: see text].

Groups of automorphisms of local fields of period p and nilpotent class < p, I

Suppose [Formula: see text] is a finite field extension of [Formula: see text] containing a primitive [Formula: see text]th root of unity. Let [Formula: see text] be a maximal [Formula: see text]-extension of [Formula: see text] with the Galois group of period [Formula: see text] and nilpotent class [Formula: see text]. In this paper, we develop formalism which allows us to study the structure of [Formula: see text] via methods of Lie theory. In particular, we introduce an explicit construction of a Lie [Formula: see text]-algebra [Formula: see text] and an identification [Formula: see text], where [Formula: see text] is a [Formula: see text]-group obtained from the elements of [Formula: see text] via the Campbell–Hausdorff composition law. In the next paper, we apply this formalism to describe the ramification filtration [Formula: see text] and an explicit form of the Demushkin relation for [Formula: see text].

ON THE -LENGTH AND THE WIELANDT LENGTH OF A FINITE -SOLUBLE GROUP

The $p$-length of a finite $p$-soluble group is an important invariant parameter. The well-known Hall–Higman $p$-length theorem states that the $p$-length of a $p$-soluble group is bounded above by the nilpotent class of its Sylow $p$-subgroups. In this paper, we improve this result by giving a better estimation on the $p$-length of a $p$-soluble group in terms of other invariant parameters of its Sylow $p$-subgroups.

COMMUTATIVE AUTOMORPHIC LOOPS OF ORDER p3

A loop is said to be automorphic if its inner mappings are automorphisms. For a prime p, denote by [Formula: see text] the class of all 2-generated commutative automorphic loops Q possessing a central subloop Z ≅ ℤp such that Q/Z ≅ ℤp × ℤp. Upon describing the free 2-generated nilpotent class two commutative automorphic loop and the free 2-generated nilpotent class two commutative automorphic p-loop Fp in the variety of loops whose elements have order dividing p2 and whose associators have order dividing p, we show that every loop of [Formula: see text] is a quotient of Fp by a central subloop of order p3. The automorphism group of Fp induces an action of GL 2(p) on the three-dimensional subspaces of Z(Fp) ≅ (ℤp)4. The orbits of this action are in one-to-one correspondence with the isomorphism classes of loops from [Formula: see text]. We describe the orbits, and hence we classify the loops of [Formula: see text] up to isomorphism. It is known that every commutative automorphic p-loop is nilpotent when p is odd, and that there is a unique commutative automorphic loop of order 8 with trivial center. Knowing [Formula: see text] up to isomorphism, we easily obtain a classification of commutative automorphic loops of order p3. There are precisely seven commutative automorphic loops of order p3 for every prime p, including the three abelian groups of order p3.