let $f_q d_{2n}$ be the group algebra of $d_{2n}$, the dihedral group of order $2n$ over $f_q=gf(q)$. in this paper, we establish the structure of $u(f_{2^k}d_{2n})$, the unit group of $f_{2^k}d_{2n}$ and that of its normalized unitary subgroup $v_*(f_{2^k}d_{2n})$ with respect to canonical involution $*$ when $n$ is odd.

g∈G agg −1 is an antiautomorphism of KG of order 2. An element v of V (KG) satisfying v = v is called unitary. We denote by V∗(KG) the subgroup of V (KG) formed by the unitary elements of KG. Let char(K) be the characteristic of the field K. In [2], A.Bovdi and A. Szákacs construct a basis for V∗(KG) where char(K) > 2. Also A. Bovdi and L. Erdei [1] determine the structure of V∗(F2G) for all gr...

Let p be a prime, K a field of characteristic p , G a locally finite p-group, KG the group algebra, and V the group of the units of KG with augmentation 1. The anti-automorphism g 7→ g of G extends linearly to KG ; this extension leaves V setwise invariant, and its restriction to V followed by v 7→ v gives an automorphism of V . The elements of V fixed by this automorphism are called unitary; t...

the unitary cayley graph xn has vertex set zn = {0, 1,…, n-1} and vertices u and v areadjacent, if gcd(uv, n) = 1. in [a. ilić, the energy of unitary cayley graphs, linear algebraappl. 431 (2009) 1881–1889], the energy of unitary cayley graphs is computed. in this paperthe wiener and hyperwiener index of xn is computed.

We show that the units found in torsion-free group rings by Gardam are twisted unitary elements. This justifies some choices Gardam's construction might have appeared arbitrary, and yields more examples of units. note all up to date exhibit non-trivial symmetry.

The unitary Cayley graph Xn has vertex set Zn = {0, 1,…, n-1} and vertices u and v are adjacent, if gcd(uv, n) = 1. In [A. Ilić, The energy of unitary Cayley graphs, Linear Algebra Appl. 431 (2009) 1881–1889], the energy of unitary Cayley graphs is computed. In this paper the Wiener and hyperWiener index of Xn is computed.