Let G be the group ({1, ı, −1, −ı}, ∙) and let H ≤ (+_− 1, ∙), show that θ:G→H is an Isomorphism. Hello!


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Question Number 192746 by Mastermind last updated on 26/May/23
$Let G be the group ({1, ı, −1, −ı}, ∙) and let H ≤ (+_− 1, ∙), show that θ:G→H is an Isomorphism. Hello!$
$Let G be the group ({1, ı, - 1, - ı}, \cdot)$ $and let H ⩽ (\underline{+} 1, \cdot), show that$ $θ : G \to H is an Isomorphism .$ $Hello!$
	Answered by aleks041103 last updated on 27/May/23
	$If I understood the question correctly, we need to show that G is iomorphic to a subgroup of the group ({+1,−1},∙). But G≅C_4 and ({+1,−1},∙)≅C_2 . ⇒∀H≤C_2 , ∣H∣∈{1,2} while ∣G∣=4 if H≅G, then ∣H∣=∣G∣=4∉{1,2} what you′re asking is not true and therefore impossible to prove.$
	$I f I u n d e r s t o o d t h e q u e s t i o n c o r r e c t l y, w e$ $n e e d t o s h o w t h a t G i s i o m o r p h i c t o a s u b g r o u p$ $o f t h e g r o u p ({+ 1, - 1}, \cdot) .$ $B u t G ≅ C_{4} a n d ({+ 1, - 1}, \cdot) ≅ C_{2} .$ $\Rightarrow \forall H \leq C_{2}, ∣ H ∣\in {1, 2}$ $w h i l e ∣ G ∣= 4$ $i f H ≅ G, t h e n ∣ H ∣=∣ G ∣= 4 \notin {1, 2}$ $w h a t {y o u}^{'} r e a s k i n g i s n o t t r u e a n d t h e r e f o r e$ $i m p o s s i b l e t o p r o v e .$
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