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?

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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?

Biardiask3zd · Accepted Answer

Step 1Yes. 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 2and 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.

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}⟩.

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