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Abstract Algebra?
Show that (ZxZ)/<(1,1)> is an infinite cyclic group.
Show that (ZxZ)/<(2,2)> is not a cyclic group.
I really don't understand modulo as far as groups go. Any help would be appreciated. Thanks.
Or I guess it's called factor (or quotient) groups. Still confused regardless.
1 Answer
- Curt MonashLv 71 decade agoFavorite Answer
1) (1,0) is a generator, because (a,b) is that generator to the a-b power.
2) (1,1) can only be a power of elements of the form (a,a). (Why?). But the set of such elements is very small -- it's just the identity and (1,1). And (1,1)^2 = identity. So obviously (1,1) doesn't generate the whole group.
