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dbrun
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Homework Statement
In Z4 x Z4, find two subgroups H and K of order 4 such that H is not isomorphic to K, but (Z4 x Z4)/H isomorphic (Z4 x Z4)/K
Homework Equations
The Attempt at a Solution
I know (Z4 x Z4) has twelve elements (0,0), (1,0), (2,0), (3,0), etc. I can generate subgroups of order 4 by picking an (a,b) in which one of two elements has order 4. For example <(2,1)> = {(2,1), (0,2), (2,3),(0,0)}, call this H. The problem is that anytime I do this, I end up with subgroups which are isomorphic b/c they each have two elements of order 4, one elt of order 2 and the identity.
So my thought was to use one of the subgroups above and then find a different subgroup of order 4 which is not isomorphic. The group I found is {(0,0), (2,0), (0,2), (2,2)}, call this K.
Now I have two subgroups which are not isomorphic, but when I figure out (Z4 x Z4)/H and (Z4 x Z4)/K I end up with quotient groups which I do not think are isomorphic.
(Z4 x Z4)/H = {H, (1,0) + H, (2,0) + H, (3, 0) + H}
(Z4 x Z4)/K = {K, (0,1) + K, (1,0) + K, (1,1) + K}
(Z4 x Z4)/H has 3 elements of order 3 and identity
(Z4 x Z4)/K has 3 elements of order 2 and identity
Am I calculating the order of these elements incorrectly?
Any suggestions or insight would be greatly appreciated.
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