The power graph ΓG of a finite group G is the graph with the vertex set G, where two distinct elements are adjacent if one is a power of the other. An L(2, 1)-labeling of a graph Γ is an assignment of labels from nonnegative integers to all vertices of Γ such that vertices at distance two get different labels and adjacent vertices get labels that are at least 2 apart. The lambda number of Γ, de...

The representation theory of the group U (1, q) is discussed in detail because of its possible application in a quaternion version of the Salam-Weinberg theory. As a consequence, from purely group theoretical arguments we demonstrate that the eigenvalues must be right-eigenvalues and that the only consistent scalar products are the complex ones. We also define an explicit quaternion tensor prod...

This paper deals with Euclidean space codes obtained from two-level group codes of length n over a Generalized Quaternion Group Q,, m23, with 2"' elements, with generators x and y, given by the presentation Q,,,= using a matched labelling of a 4-dimensional signal set with 2" elements with Q,. I. TWO LEVEL GROUP CODES OVER e,. Four dimensional signal sets in conne...

We study the inequities in distribution of Frobenius elements Galois extensions rational numbers with groups that are either dihedral $D_{2n}$ or (generalized) quaternion $\mathbb H_{2n}$ two-power order. In spirit recent work Fiorilli and Jouve arXiv:2001.05428, we study, under natural hypotheses, some families such extensions, a horizontal aspect, where degree is fixed, vertical goes to infin...

We show that an action of SL(2, p), p ≥ 7 an odd prime such that 4 6 | (p − 1), has exactly two orbital digraphs Γ1, Γ2, such that Aut(Γi) admits a complete block system B of p + 1 blocks of size 2, i = 1, 2, with the following properties: the action of Aut(Γi) on the blocks of B is nonsolvable, doubly-transitive, but not a symmetric group, and the subgroup of Aut(Γi) that fixes each block of B...

A length n group code over a group G is a subgroup of G under component-wise group operation. Two-level group codes over the class of generalized quaternion groups, Q2m, TM > 3, are constructed using a binary code and a code over Z2m-i, the ring of integers modulo 2 , as component codes and a mapping / from Z2 X Z2m_i to Q2m. A set of necessary and sufficient conditions on the component codes i...