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May 16th, 2012, 12:08 AM  #1 
Senior Member Joined: Apr 2012 Posts: 106 Thanks: 0  Show that group Z2 x Z2 is not isomorphic to the group Z4
Hello dear colleagues, I need to show that the group Z2 X Z2 is not isomorphic to group Z4. (Z is a whole number). I would appreciate any offered help, V 
May 16th, 2012, 06:31 AM  #2 
Senior Member Joined: Apr 2012 Posts: 106 Thanks: 0  Re: Show that group Z2 x Z2 is not isomorphic to the group Z
I am completely new at this field  so I don't even get how do we get order from the group Z2 x Z2. For example, whats the order of (1,0)? How do we calculate that? 
May 16th, 2012, 12:58 PM  #3 
Senior Member Joined: Jul 2011 Posts: 227 Thanks: 0  Re: Show that group Z2 x Z2 is not isomorphic to the group Z
The order of The order of a group is the cardinality of the group. Do you know the definition of the order of an element of a group? To solve the exercice I think you need to find a function which is not bijective. 
May 16th, 2012, 01:37 PM  #4 
Senior Member Joined: Apr 2012 Posts: 106 Thanks: 0  Re: Show that group Z2 x Z2 is not isomorphic to the group Z
I used the rule that order of elements in Z2 x Z2 must be equal to the order of elements in Z4. It worked.

May 17th, 2012, 05:08 PM  #5 
Senior Member Joined: Mar 2012 Posts: 294 Thanks: 88  Re: Show that group Z2 x Z2 is not isomorphic to the group Z
the standard proof goes something like this: (0,1) in Z2 x Z2 has order 2: (0,1) + (0,1) = (0,0) (since 1+1 = 2 = 0 (mod 2)). (1,0) in Z2 x Z2 also has order 2: (1,0) + (1,0) = (0,0). now if Z2 x Z2 and Z4 were isomorphic, Z4 would contain (at least) 2 elements of order 2 (the images of (1,0) and (0,1) under the isomorphism). but: 0 is of order 1 1+1 = 2 1+1+1 = 3 1+1+1+1 = 0 <1 is of order 4 2+2 = 0 <2 is of order 2 3+3 = 2 (since 6 = 2 (mod 4)) 3+3+3 = 1 (since 9 = 1 (mod 4)) 3+3+3+3 = 0 <3 is of order 4. since Z4 has only one element of order 2 (namely, 2), it cannot be isomorphic to Z2 x Z2. 
May 28th, 2012, 10:16 AM  #6 
Newbie Joined: May 2012 Posts: 3 Thanks: 0  Re: Show that group Z2 x Z2 is not isomorphic to the group Z dosn't have an element of order 4 (, ). But does (for example, ). So, they are nonisomorphic. P.S.:  order of the element .

June 5th, 2012, 02:58 PM  #7 
Math Team Joined: Sep 2007 Posts: 2,409 Thanks: 5  Re: Show that group Z2 x Z2 is not isomorphic to the group Z
To me the most obvious point is that [itex]Z_2\times Z_2[/itex] has the property that a+ a= 0 for all a. That is not true for [itex]Z_4[/itex].


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