(1) Prove that O(n) is homeomorphic to S O ( n ) × Z 2 . (2) Are these two isomorphic as topological groups?(O(n) : Orthogonal group, SO(n) : Special orthogonal group)My attempt for (1) :Let's pick any element C from O ( n ) − S O ( n ), and let f : S O ( n ) → S O ( n ) ; f ( A ) = A g : O ( n ) − S O ( n ) → S O ( n ) ; g ( A ) = C A Then f and g are homeomorphisms. Since SO(n) is compact, g − 1 ( S O ( n ) ) = O ( n ) − S O ( n ) is also compact. So, they are closed in the subspace topology on O(n). Thus, by the glueing lemma, the function φ : O ( n ) → S O ( n ) × Z 2 ; φ ( A ) = { ( f ( A ) , 0 ) ( when A ∈ S O ( n ) ) ( g ( A ) , 1 ) ( otherwise ) becomes continuous. We can easily confirm that φ is bijective and φ − 1 is continuous.Questions:(1) I'm not sure I'm going in the right direction (in proving (1)). I used glueing lemma to justifying continuousness of the function φ, but I doubt whether I'm using that lemma in the right place.(2) I tried to prove that two groups are NOT isomorphic, but I can't find any key for that. I tried to compare orders of elements in two groups, but I couldn't find some good elements to compare... Is there any simple way to check whether two groups are isomorphic?

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(1) Prove that O(n) is homeomorphic to       S    O    (  n  )  ×            Z        2  . (2) Are these two isomorphic as topological groups?(O(n) : Orthogonal group, SO(n) : Special orthogonal group)My attempt for (1) :Let&#039;s pick any element C from       O    (  n  )  −      S    O    (  n  ), and let                          f        :                  S          O                (        n        )        →                  S          O                (        n        )        ;                f        (        A        )        =        A                                  g        :                  O                (        n        )        −                  S          O                (        n        )        →                  S          O                (        n        )        ;                g        (        A        )        =        C        A            Then f and g are homeomorphisms. Since SO(n) is compact,       g          −      1        (      S    O    (  n  )  )  =      O    (  n  )  −      S    O    (  n  ) is also compact. So, they are closed in the subspace topology on O(n). Thus, by the glueing lemma, the function  φ  :      O    (  n  )  →      S    O    (  n  )  ×            Z        2    ;    φ  (  A  )  =      {                            (          f          (          A          )          ,          0          )                                    (          when           A          ∈                      S            O                    (          n          )          )                                      (          g          (          A          )          ,          1          )                          (          otherwise          )                        becomes continuous. We can easily confirm that   φ is bijective and       φ          −      1       is continuous.Questions:(1) I&#039;m not sure I&#039;m going in the right direction (in proving (1)). I used glueing lemma to justifying continuousness of the function   φ, but I doubt whether I&#039;m using that lemma in the right place.(2) I tried to prove that two groups are NOT isomorphic, but I can&#039;t find any key for that. I tried to compare orders of elements in two groups, but I couldn&#039;t find some good elements to compare... Is there any simple way to check whether two groups are isomorphic?

antidootnw · Accepted Answer

Step 1Consider the mapping   f  :  O  (  n  )  →  S  O  (  n  )  ×            Z        2   given by  K  →  {  det  (  M  )  ⋅  M  ,  det  M  }This is a well defined one to one and onto map. You can check that the map above is continuous.Step 2Now since we know that O(n) is compact and   S  O  (  n  )  ×            Z        2   is Hausdorff , the following map is a homeomorphism.

Riya Andrews · Accepted Answer

Step 1Consider the mapping   f  :  O  (  n  )  →  S  O  (  n  )  ×      Z    2    2 given by   M  →  (  d  e  t  (  M  )  ⋅  M  ,  d  e  t  M  )Step 2I think your answer is not right. Check det   [  d  e  t  (  M  )  M  ]  =  [  d  e  t  (  M  )      ]          n      +      1      . Thus if n is even, it will be wrong.

(1) Prove that O(n) is homeomorphic to SO(n) times Z_2. (2) Are these two isomorphic as topological groups?

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