The Intersection Graph of Subgroups of the Dihedral Group of Order 2pq

For a finite group G, the intersection graph of G is the graph whose vertex set is the set of all proper non-trivial subgroups of G, where two distinct vertices are adjacent if their intersection is a non-trivial subgroup of G. In this article, we investigate the detour index, eccentric connectivity, and total eccentricity polynomials of the intersection graph of subgroups of the dihedral group for distinct primes . We also find the mean distance of the graph .

On the dimension of $H^{*}((\mathbb Z_2)^{\times t}, \mathbb Z_2)$ as a module over Steenrod ring

We write $\mathbb P$ for the polynomial algebra in one variable over the finite field $\mathbb Z_2$ and $\mathbb P^{\otimes t} = \mathbb Z_2[x_1, \ldots, x_t]$ for its $t$-fold tensor product with itself. We grade $\mathbb P^{\otimes t}$ by assigning degree $1$ to each generator. We are interested in determining a minimal set of generators for the ring of invariants $(\mathbb P^{\otimes t})^{G_t}$ as a module over Steenrod ring, $\mathscr A_2.$ Here $G_t$ is a subgroup of the general linear group $GL(t, \mathbb Z_2).$ An equivalent problem is to find a monomial basis of the space of "unhit" elements, $\mathbb Z_2\otimes_{\mathscr A_2} (\mathbb P^{\otimes t})^{G_t}$ in each $t$ and degree $n\geq 0.$ The structure of this tensor product is proved surprisingly difficult and has been not yet known for $t\geq 5,$ even for the trivial subgroup $G_t = \{e\}.$ In the present paper, we consider the subgroup $G_t = \{e\}$ for $t \in \{5, 6\},$ and obtain some new results on $\mathscr A_2$-generators of $(\mathbb P^{\otimes t})^{G_t}$ in some degrees. At the same time, some of their applications have been proposed. We also provide an algorithm in MAGMA for verifying the results. This study can be understood as a continuation of our recent works in [23, 25].

On the dimension of $H^{*}((\mathbb Z_2)^{\times t}, \mathbb Z_2)$ as a module over Steenrod ring

We write $\mathbb P$ for the polynomial algebra in one variable over the finite field $\mathbb Z_2$ and $\mathbb P^{\otimes t} = \mathbb Z_2[x_1, \ldots, x_t]$ for its $t$-fold tensor product with itself. We grade $\mathbb P^{\otimes t}$ by assigning degree $1$ to each generator. We are interested in determining a minimal set of generators for the ring of invariants $(\mathbb P^{\otimes t})^{G_t}$ as a module over Steenrod ring, $\mathscr A_2.$ Here $G_t$ is a subgroup of the general linear group $GL(t, \mathbb Z_2).$ Equivalently, we want to find a basis of the $\mathbb Z_2$-vector space $\mathbb Z_2\otimes_{\mathscr A_2} (\mathbb P^{\otimes t})^{G_t}$ in each degree $n\geq 0.$ The problem is proved surprisingly difficult and has been not yet known for $t\geq 5.$ In the present paper, we consider the trivial subgroup $G_t = \{e\}$ for $t \in \{5, 6\},$ and obtain some new results on $\mathscr A_2$-generators for $(\mathbb P^{\otimes 5})^{G_5}$ in degree $5(2^{1} - 1) + 13.2^{1}$ and for $(\mathbb P^{\otimes 6})^{G_6}$ in "generic" degree $n = 5(2^{d+4}-1) + 47.2^{d+4}$ with a positive integer $d.$ An efficient approach to studying $(\mathbb P^{\otimes 5})^{G_5}$ in this case has been provided. In addition, we introduce an algorithm on the MAGMA computer algebra for the calculation of this space. This study is a continuation of our recent works in \cite{D.P2, D.P4}.

The C*–algebras of Compact Transformation Groups

We investigate the representation theory of the crossed–product C*–algebra associated with a compact group G acting on a locally compact space X when the stability subgroups vary discontinuously. Our main result applies when G has a principal stability subgroup or X is locally of finite G–orbit type. Then the upper multiplicity of the representation of the crossed product induced from an irreducible representation V of a stability subgroup is obtained by restricting V to a certain closed subgroup of the stability subgroup and taking the maximum of the multiplicities of the irreducible summands occurring in the restriction of V. As a corollary we obtain that when the trivial subgroup is a principal stability subgroup; the crossed product is a Fell algebra if and only if every stability subgroup is abelian. A second corollary is that the C*–algebra of the motion group ℝn ⋊ SO(n) is a Fell algebra. This uses the classical branching theorem for the special orthogonal group SO(n) with respect to SO(n − 1). Since proper transformation groups are locally induced from the actions of compact groups, we describe how some of our results can be extended to transformation groups that are locally proper.

Class preserving automorphisms of Blackburn groups

In this article, a Blackburn group refers to a finite non-Dedekind group for which the intersection of all nonnormal subgroups is not the trivial subgroup. By completing the arguments of M. Hertweck, we show that all conjugacy class preserving automorphisms of Blackburn groups are inner automorphisms.

The structure of an infinite Sylow subgroup in some periodic Shunkov groups

We study periodic groups such that the normaliser of any finite non-trivial subgroup of such a group is almost layer-finite. The class of groups satisfying this condition is rather wide and includes the free Burnside groups of odd period which is greater than 665 and the groups constructed by A. Yu. Olshanskii.We consider the classical question: how the properties of the system of subgroups of a group influence on the properties of the group? We show that almost layer-finiteness is transferred on the group G from the normalisers of non-trivial finite subgroups of the group G if G is periodic conjugately biprimitively finite group with a strongly embedded subgroup.We study the structure of an infinite Sylow 2-subgroup in a periodic conjugately biprimitively finite group in the case that the normaliser of any finite non-trivial subgroup is almost layer-finite.The results of the paper can be useful in the study of the class of periodic conjugately biprimitively finite groups (periodic Shunkov groups).

Tameness and geodesic cores of subgroups

Let N be a finitely generated normal subgroup of a finitely generated group G. We show that if the trivial subgroup is tame in the factor group G/N, then N is that in G. We also give a short new proof of the fact that quasiconvex subgroups of negatively curved groups are tame. The proof utilizes the concept of the geodesic core of the subgroup and is related to the Dehn algorithm.

A note on the fundamental group of a manifold of negative curvature

Let Y be a compact connected C∞ Riemannian manifold with negative sectional curvatures. Let G be a non-trivial subgroup of the fundamental group π1(Y). G is known to be cyclic if it is abelian (Preissmann (6)) or contains a subnormal abelian (hence cyclic) subgroup (Yau(9)). These results may be generalized as follows: Say that a group G is of type (α) if ∃a ∈ G, a ≠ e, such that for all b belonging to a set of generators for G we have ambn = bqap for some integers m, n, p, q with either m = p or n = q.

Highly Symmetric Homogeneous Spaces

We consider effective homogeneous spacesM=G/HwhereGis a compact connected Lie group,His a closed subgroup andGacts effectively onM(i.e.,Hcontains no non-trivial subgroup normal inG). It is known that dimG≦m2/2 +m/2 wherem= dimMand that if dimG=m2/2 +m/2, thenMis diffeomorphic to the standard sphereSmor the standard real projective spaceRPm[1]. In addition it has been shown that for fixedmthere are gaps in the possible dimensions forGbelow the maximal bound [4; 5].

Recurrent Transformation Groups

Let (X, T, π) denote a flow, whereXis a compact topological space metrizable byd, andTis a closed non-trivial subgroup of the reals under addition.Tisrecurrentif and only if for eachands> 0, there existst>ssuch thatx∈Ximplies. IfTis almost-periodic, thenTis both recurrent and distal. In§§4 and 5, it is shown that, under more stringent hypotheses, the recurrence ofTis neither a necessary nor a sufficient condition forTto be distal. LetSbe a closed non-trivial subgroup ofT. It is shown in§3 thatTis recurrent if and only ifSis recurrent. From this result, we obtain a solution to a problem posed by Nemyckiĭ (16, p. 492, Problem 6).