On the existence of the orthogonal basis of the symmetry classes of tensors associated with certain groups

On distinguishing labelling of groups for the conjugation action

“…Later, Darafsheh and Poursalavati [2], observed that with the above presentation, the group V 8n can also be defined for an arbitrary n. However, the conjugacy classes of the group V 8n differ, depending upon whether n is an even or an odd positive integer. When n is odd, the group V 8n has 2n + 3 conjugacy classes precisely;…”

“…On the other hand, if g = ba i ; 0 ≤ i ≤ 4n − 1, then in this case, fix h = a 2n−1 ∈ X ′ n , which leads to ghg −1 = a i ba 2n−1 b −1 a −i = a i aa −i = a and certainly ghg −1 = a / ∈ X ′ n . Thus {g ∈ SD 8n : ϕ(ghg −1 ) = ϕ(h), for all h ∈ SD 8n } ⊆ {1, a 2n } and the reverse inclusion follows by using Lemma 5.3 (2). ■…”