Relation between the left and right cosets of an α-normal subgroup

Abstract

Let G be a group and α be an automorphism of G. In 2016, Ganjali and Erfanian introduced the notion of a normal subgroup related to α, called the α-normal subgroup. It is basically known that if N is an ordinary normal subgroup of G then every right coset Ng is actually the left coset gN. This fact allows us to define the product of two right cosets naturally, thus inducing the quotient group. This research investigates the relation between the left and right cosets of the relative normal subgroup. As we have done in the classic version, we then define the product of two right cosets in a natural way and continue with the construction of a, say, relative quotient group.

A Note on Relative Normal Subgroups

“…Let 𝐺 be a group and 𝑁 be a nontrivial subgroup of 𝐺 such that 𝑁 is 𝛼-normal in 𝐺 for an 𝛼 ∈ 𝐴𝑢𝑡(𝐺). It was shown in Mahatma, et al, (2021) that, for every ℎ ∈ 𝐺, ℎ𝑁 = 𝑁𝛼(ℎ). Now suppose that 𝜏 ∈ 𝐴𝑢𝑡(𝐺) such that 𝑁 is 𝜏-normal in 𝐺.…”

“…Thus, 𝜏(𝑔) = 𝑔𝑛′ and hence, for every 𝑛 ∈ 𝑁, 𝑔 −1 𝑛𝛼𝜏(𝑔) = 𝑔 −1 𝑛𝛼(𝑔𝑛′) = 𝑔 −1 𝑛𝛼(𝑔)𝛼(𝑛′). Now, it was shown in Mahatma, et al, (2021) that 𝛼 must satisfy 𝛼(𝑁) = 𝑁 and thus, 𝛼(𝑛′) ∈ 𝑁. Next, since 𝑁 is 𝛼-normal then 𝑔 −1 𝑛𝛼(𝑔) ∈ 𝑁 and thus, we have 𝑔 −1 𝑛𝛼𝜏(𝑔) ∈ 𝑁.…”

“…al., (2023. Recently, Mahatma, et al, (2021) formulated the basic properties of such subgroups and obtain some results analogous to the classic version. They showed that if 𝐺 is a group and 𝛼 ∈ 𝐴𝑢𝑡(𝐺) then the followings are equivalent:…”