Two finite groups are to be compared in terms of structure (whether isomorphic or not): ⟨x,y∣x^2, y^2, (xy)^2⟩ and ⟨x,y∣x^4, y^2=x^2, yxy^{-1}=x^{-1}⟩.

Joglxym

Joglxym

Answered question

2022-11-10

Argument that two given finite groups are not isomorphic.
Two finite groups are to be compared in terms of structure (whether isomorphic or not): x , y x 2 ,   y 2 ,   ( x y ) 2 and x , y x 4 ,   y 2 = x 2 ,   y x y 1 = x 1 .
First group is very simple, the second one not so much. However I found both of them on Wikipedia - their orders are 4 and 8 respectively.
So for the second group do I still have to figure out all the relations (to arrive at order 8) just to make a statement that orders are not equal hence no isomorphism? Is there a quick formal argument for example that order is greater than 4?

Answer & Explanation

Biardiask3zd

Biardiask3zd

Beginner2022-11-11Added 16 answers

Step 1
Yes. That presentation is recognized as the presentation of a Dicyclic group. For reference, the dicyclic group of degree n is given by
D i c n = a , x a 2 n = 1 ,   x 2 = a n ,   x a x 1 = a 1
Step 2
and has order 4n. Your group is just D i c 2 and so has order 8. By the way, D i c 2 Q 8 , the quaternion group.

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