We demonstrate that two supersoluble complements of an abelian base in a finite split extension are conjugate if and only if, for each prime $p$, Sylow $p$-subgroup one complement is to the other. As corollary, we find any subgroup $N$ $G$ there exists $S$ such $S\cap N$ $G$. In particular, restricting groups allows us ease D. G. Higman's stipulation be within $S$. then consider group actions o...

We propose a new description of the SU(N) Yang-Mills theory on a lattice, which enables one to explain quark confinement based on the dual superconductivity picture in a gauge independent way. This is because we can define gauge-invariant magnetic monopoles which are inherent in the Wilson loop operator. For SU(3) there are two options: the minimal option with a single type of non-Abelian magne...

The security of two public key encryption schemes relying on the hardness of different computational problems in non-abelian groups is investigated. First, an attack on a conceptual public key scheme based on Grigorchuk groups is presented: We show that from the public data one can easily derive an ‘equivalent’ secret key that allows the decryption of arbitrary messages encrypted under the publ...

The energy density is computed for a U(2) Chern-Simons theory coupled to a nonrelativistic fermion field (a theory of “non-Abelian anyons”) under the assumptions of uniform charge and matter density. When the matter field is a spinless fermion, we find that this energy is independent of the two Chern-Simons coupling constants and is minimized when the non-Abelian charge density is zero. This su...

A group $G$ has a finite special rank $r$ if every finitely generated subgroup of is by at most elements and there which exactly generators. If not such $r$, then we say that infinite rank. In this paper, study generalized radical non-abelian groups whose subgroups are transitively normal.

We show that the outer automorphism groups of graph products finitely generated abelian satisfy Tits alternative, are residually finite, their so-called Torelli subgroups generated, and they a dichotomy between being virtually nilpotent containing non-abelian free subgroup is determined by graphical condition on underlying labelled graph. Graph simultaneously generalize right-angled Artin (RAAG...

In this short note, we show that the class of abelian groups determined by the subgroup lattice of their direct n-powers is exactly the class of the abelian groups which share the n-root property. As applications we answer in the negative a (semi)conjecture of Palfy and solve a more general problem. Recently, for an arbitrary group G, the subgroup lattice of the square G×G has received some att...

Comfort, W.W. and J. van Mill, Some topological groups with, and some without, proper dense subgroups, Topology and its Applications 41 (1991) 3-15. Continuing earlier investigations into the question cf the existence of a proper dense subgroup of a given topological group, the authors obtain some positive and some negative results, as follows: (a) Every non-degenerate connected Abelian group c...

The totality formed by all the operators of any group (G) which are common to all the invariant subgroups of prime index (p) constitutes a characteristic subgroup, and the corresponding quotient group is the abelian group of order pK and of type (1, 1, 1, ■■■)-\ The number of the invariant subgroups of index p is therefore pK — 1/p — 1. The given totality includes all the operators of G which a...

The totality formed by all the operators of any group (G) which are common to all the invariant subgroups of prime index (p) constitutes a characteristic subgroup, and the corresponding quotient group is the abelian group of order pK and of type (1, 1, 1, ■■■)-\ The number of the invariant subgroups of index p is therefore pK — 1/p — 1. The given totality includes all the operators of G which a...