در ﺑﺎب 𝒏𝒔𝒆 ﺑﻌﻀﯽ ﮔﺮوه ﻫﺎي ﺧﺎص

on the nse of some particular groups

ﺑﺮاي ﮔﺮوه G، ﮔﯿﺮﯾﻢ )nse(G ﻧﺸﺎن دﻫﻨﺪه ﻣﺠﻤﻮﻋﻪ ﺗﻌﺪاد ﻋﻨﺎﺻﺮ از ﻣﺮﺗﺒﻪ ﻣﺸﺎﺑﻪ در G ﺑﺎﺷﺪ. در اﯾﻦ ﻣﻘﺎﻟﻪ، ﻧﺸﺎن ﻣﯽ دﻫﯿﻢ ﺑﺮاي ﯾﮏ ﮔﺮوه ﻣﺘﻨﺎﻫﯽ و ﻓﺮوﺑﻨﯿﻮس G ﺑﺎ ﺳﺎﺧﺘﺎر ﻣﺸﺨﺺ و ﮔﺮوه دﻟﺨﻮاه L، اﮔﺮ )nse(G)=nse(L، آﻧﮕﺎه 𝐺 ≅ 𝐿. ﻫﻤﭽﻨﯿﻦ ﺑﺎ اﺳﺘﻔﺎده از ﻋﻨﺎﺻﺮ ﻣﺠﻤﻮﻋﻪ nse، ﻣﺤﮏ ﺟﺪﯾﺪي ﺑﺮاي ﺗﺸﺨﯿﺺ ﮔﺮوه ﻫﺎي ﭘﻮچ ﺗﻮان اراﺋﻪ ﻣﯽ ﮔﺮدد.

For a finite group $ G $, let $ { m n s e } ( G ) $ be the set of the number of the elements of the same order in $ G $. In this paper, we first study the set $ n s e $ of a Frobenius group , the set $ { m n s e } $ of a $ 2 $- Frobenius group and the set $ { m n s e } $ of a nilpotent group. Then, we show that for the finite non-solvable Frobenius group $ G $ with the certain structure and an arbitrary group $ L $ , if $ { m n s e } ( G ) = { m n s e } ( L ) $, then $ G ≅ L $. Also, a new criterion is presented to recognize nilpotent groups by their $ { m n s e }$.