Let $G$ be a $p$-group of order $p^n$ and $Phi$=$Phi(G)$ be the Frattini subgroup of $G$. It is shown that the nilpotency class of $Autf(G)$, the group of all automorphisms of $G$ centralizing $G/ Fr(G)$, takes the maximum value $n-2$ if and only if $G$ is of maximal class. We also determine the nilpotency class of $Autf(G)$ when $G$ is a finite abelian $p$-group.

Let A be a subset of an abelian group G with |G| = n. We say that A is sum-free if there do not exist x, y, z ∈ A with x+ y = z. We determine, for any G, the maximal density μ(G) of a sum-free subset of G. This was previously known only for certain G. We prove that the number of sum-free subsets of G is 2, which is tight up to the o-term. For certain groups, those with a small prime factor of t...

Abstract Let A ⊆ M ⊆ B(L2(M)) be a maximal abelian self-adjoint subalgebra (masa) in a type II1 factor M in its standard representation. The abelian von Neumann algebra A generated by A and JAJ has a type I commutant which contains the projection eA ∈ A onto L2(A). Then A′(1− eA) decomposes into a direct sum of type In algebras for n ∈ {1, 2, · · · ,∞}, and those n’s which occur in the direct s...

In this paper we associate a semigroup to a locally maximal subset of complete controllability, i.e., a local control set. This fundamental semigroup is based on equivalence classes under homotopies in the set of trajectories. It reflects the structure of the set of closed (trajectory) loops in the local control set. We discuss the relations between different local control sets and prove a Van ...

The main purpose of this note is to observe that the celebrated theorem of Malliavin on the failure of spectral synthesis for the group algebra L{G) of a non compact locally compact abelian group G can be strengthened to get the failure of weak spectral synthesis (in the sense of Warner [7]) for £(G) and also for the "generalised group algebra" L{G, A). Let A be a commutative, semisimple, regul...

Filters of a lattice, maximal filters (ultrafilters), operation to create a filter generating by an element or by a non-empty subset of the lattice are discussed. Besides, there are introduced implicative lattices such that for every two elements there is an element being pseudo-complement of them. Some facts concerning these concepts are presented too, i.e. for any proper filter there exists a...

Let G be a finite group and S a subset of G. Then S is product-free if S ∩ SS = ∅, and complete if G∗ ⊆ S ∪ SS. A product-free set is locally maximal if it is not contained in a strictly larger product-free set. If S is product-free and complete then S is locally maximal, but the converse does not necessarily hold. Street and Whitehead [11] defined a group G as filled if every locally maximal p...

the full automorphism group of $u_6(2)$ is a group of the form $u_6(2){:}s_3$. the group $u_6(2){:}s_3$ has a maximal subgroup $2^9{:}(l_3(4){:}s_3)$ of order 61931520. in the present paper, we determine the fischer-clifford matrices (which are not known yet) and hence compute the character table of the split extension $2^9{:}(l_3(4){:}s_3)$.

The orthogonal simple group 0 (3) has three conjugacy classes of maximal subgroups the form 36:L4(3). These groups are all isomorphic to each other and order 4421589120 with index 1120 in (3). In this paper, we will compute ordinary carácter table one these using technique Fischer-Clifford matrices. This is very efficient character an extension Ḡ = N.G especially where normal subgroup N element...