A Note on Relative Normal Subgroups

Yudi Mahatma, Ibnu Hadi, Binastya Anggara Sekti

Abstract


Let G be finite group and α∈Aut(G). In 2016, Ganjali and Erfanian introduced the notion of a normal subgroup of G which is relative to α, called the α-normal subgroup. In detail, N is an α-normal subgroup of G if for every g∈G and n∈N we have g^(-1) nα(g)∈N. In this research, we show that if N is an α-normal subgroup of G and τ∈Aut(G) then N is τ-normal in G if and only if α(g)τ(g^(-1) )∈N for every g∈G.


Keywords: Group, Normal subgroup, α-normal subgroup.


Abstrak


Misalkan G grup hingga dan α∈Aut(G). Tahun 2016, Ganjali dan Erfanian memperkenalkan ide tentang suatu subgrup normal dari G relatif terhadap α, dinamakan subgrup normal-α. Secara rinci, N adalah suatu subgrup normal-α dari G apabila untuk setiap g∈G dan n∈N berlaku g^(-1) nα(g)∈N. Dalam penelitian ini diperlihatkan bahwa jika N adalah suatu subgrup normal-α dari G dan τ∈Aut(G) maka N normal-τ di G jika dan hanya jika α(g)τ(g^(-1) )∈N untuk setiap g∈G.


Keywords


Grup, Normal Subgrup, α-Normal Subgroup.

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References


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DOI: https://doi.org/10.17509/jem.v11i1.56092

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