Is Z 84 ⊕ Z 72 isomorphic to Z 36 ⊕ Z 168 ?These two groups have the same order. Also, we cannot show that the groups are not isomorphic by comparing the order of groups' elements. Thus, it seems that these two groups are isomorphic.

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Is             Z              84        ⊕            Z              72       isomorphic to             Z              36        ⊕            Z              168      ?These two groups have the same order. Also, we cannot show that the groups are not isomorphic by comparing the order of groups&#039; elements. Thus, it seems that these two groups are isomorphic.

Kelbelol · Accepted Answer

Step 1They are indeed isomorphic, and to notice this, use the fact that             Z              m      n        =            Z        n    ×            Z        m   if and only if   gcd  (  m  ,  n  )  =  1And since   84  =  12  ⋅  7 and   72  =  8  ⋅  9 you may decompose the L.H.S intoStep 2            Z        9    ×            Z        8    ×            Z        7    ×            Z        3    ×            Z        4   and show that something similar may be done with the RHS.

trapskrumcu · Accepted Answer

Step 1More generally,             Z        m    ×            Z        n    ≅            Z              gcd      (      m      ,      n      )        ×            Z              lcm      ⁡      (      m      ,      n      )      Step 2Therefore,            Z              84        ⊕            Z              72        ≅            Z              12        ⊕            Z              504        ≅            Z              36        ⊕            Z              168      because   gcd  (  84  ,  72  )  =  12  =  gcd  (  36  ,  108  ) and   lcm  ⁡  (  84  ,  72  )  =  504  =  lcm  ⁡  (  36  ,  108  ).

Is Z_{84} ⊕ Z_{72} isomorphic to Z_{36} ⊕ Z_{168}?

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