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Let-G-be-the-group-1-1-and-let-H-1-show-that-G-H-is-an-Isomorphism-Hello-

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Question Number 192746 by Mastermind last updated on 26/May/23
Let G be the group ({1, ı, −1, −ı}, ∙) and let H ≤ (+_− 1, ∙), show that θ:G→H is an Isomorphism. Hello!
LetGbethegroup({1,ı,−1,−ı},⋅)andletH⩽(+−1,⋅),showthatθ:G→HisanIsomorphism.Hello!
Answered by aleks041103 last updated on 27/May/23
If I understood the question correctly, we need to show that G is iomorphic to a subgroup of the group ({+1,−1},∙). But G≅C_4 and ({+1,−1},∙)≅C_2 . ⇒∀H≤C_2 , ∣H∣∈{1,2} while ∣G∣=4 if H≅G, then ∣H∣=∣G∣=4∉{1,2} what you′re asking is not true and therefore impossible to prove.
IfIunderstoodthequestioncorrectly,weneedtoshowthatGisiomorphictoasubgroupofthegroup({+1,−1},⋅).ButG≅C4and({+1,−1},⋅)≅C2.⇒∀H≤C2,∣H∣∈{1,2}while∣G∣=4ifH≅G,then∣H∣=∣G∣=4∉{1,2}whatyou′reaskingisnottrueandthereforeimpossibletoprove.

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