Related papers: Characteristic Subgroups of Finite Abelian Groups

In this paper, we determine the genus of the subgroup lattice of several families of abelian groups. In doing so, we classify all finite abelian groups whose subgroup lattices can be embedded into the torus.

The lattice of fully invariant subgroups of an abelian $p$--group and the lattice of ideals of its endomorphism ring are classified by systems of cardinal invariants.

In this paper we give invariants that characterize isotypically equivalent Abelian periodic groups. Also, we describe types of standart tuples of elements in these groups. As the particular case we prove that two Abelian $p$-groups with…

One way of expressing the self-duality $A\cong \Hom(A,\mathbb{C})$ of Abelian groups is that their character tables are self-transpose (in a suitable ordering). Noncommutative groups fail to satisfy this property. In this paper we extend…

We consider the families of finite Abelian groups $\ZZ/p\ZZ\times \ZZ/p\ZZ$, $\ZZ/p^2\ZZ$ and $\ZZ/p\ZZ\times \ZZ/q\ZZ$ for $p,q$ two distinct prime numbers. For the two first families we give a simple characterization of all functions…

We consider lattices generated by finite Abelian groups. We prove that such a lattice is strongly eutactic, which means the normalized minimal vectors of the lattice form a spherical 2-design, if and only if the group is of odd order or if…

A notion of degeneration of elements in groups is introduced. It is used to parametrize the orbits in a finite abelian group under its full automorphism group by a finite distributive lattice. A pictorial description of this lattice leads…

Let $G$ be a finite non-cyclic $p$-group of order at least $p^3$. If $G$ has an abelian maximal subgroup, or if $G$ has an elementary abelian centre with $C_G(Z(\Phi(G))) \ne \Phi(G)$, then $|G|$ divides $|\text{Aut}(G)|$.

In this paper we consider finite groups G satisfying the following condition: G has two columns in its character table which differ by exactly one entry. It turns out that such groups exist and they are exactly the finite groups with a…

We consider the structure of finite $p$-groups $G$ having precisely three characteristic subgroups, namely $1$, $\Phi(G)$ and $G$. The structure of $G$ varies markedly depending on whether $G$ has exponent $p$ or $p^2$, and, in both cases,…

Let $FG$ be the group algebra of a finite $p$-group $G$ over a finite field $F$ of positive characteristic $p$. Let $\cd$ be an involution of the algebra $FG$ which is a linear extension of an anti-automorphism of the group $G$ to $FG$. If…

We prove that an isomorphism between saturated fusion systems over the same finite p-group is detected on the elementary abelian subgroups of the hyperfocal subgroup if p is odd, and on the abelian subgroups of the hyperfocal subgroup of…

In the present paper we introduce and study a canonical ${\cal E}$-lattice structure on the set of element orders of some finite groups. We show that a finite abelian group is uniquely determined by this canonical ${\cal E}$-lattice.

In this paper, we study the posets of classes of subgroups of finite group having same set of orders of elements. We show that this poset is a chain only in the case of p-groups and moreover, we characterize all finite groups for which this…

We classify, up to isomorphism, all gradings by an arbitrary abelian group on simple finitary Lie algebras of linear transformations (special linear, orthogonal and symplectic) on infinite-dimensional vector spaces over an algebraically…

In the note we prove that all composition factors of a finite group possessing a Carter subgroup of odd order either are abelain, or are isomorphic to $L_2(3^{2n+1})$.

A finite group $G$ is called a Schur group if any Schur ring over $G$ is associated in a natural way with a subgroup of $Sym(G)$ that contains all right translations. It is proved that the group $C_3\times C_3\times C_p$ is Schur for any…

We show that for non-conjugate subgroups $G_1$ and $G_2$ of a finite group $G$ there exists an extension of $G$ (by a finite group) in which the pre-images of $G_1$ and $G_2$ are not isomorphic. This allows us to show that $\mathbb Z$-coset…

It is known that any locally graded group with finitely many derived subgroups of non-normal subgroups is finite-by-abelian. This result is generalized here, by proving that in a locally graded group $G$ the subgroup $\gamma_{k}(G)$ is…

We study abstract finite groups with the property, called property $\hat{s}$, that all of their subrepresentations have submultiplicative spectra. Such groups are necessarily nilpotent and we focus on $p$-groups. $p$-groups with property…