Why are the Integers a Cyclic Group?

If we follow Wikipedia in defining a cyclic group as a group in which there exists an element g in G such that G = <g> = { gn | n is an integer }, then the integers under addition are clearly a cyclic group with the generator 1. But why do we define cyclic groups that way? Or, another way of putting it, why is the definition given the name cyclic when there’s nothing cyclic about it?

The name cyclic makes a lot more sense for finite cyclic groups in which successive additions wrap around to the identity. E.g. in ℤ4, 1 + 1 = 2, 2 + 1 = 3, 3 + 1 = 0, 0 + 1 = 1, and then you start over. It cycles with period 4.

But in the full integers you never wrap around. 1 never generates 0 no matter how many times you add it to itself. There are no cycles.

Is there some reasonable sense in which \( \sum_{k=1}^\infty 1 = \sum_{k=1}^\infty -1 \)?

I.e. is there some way of mapping the integer infinity (which some mathematicians would argue is an oxymoron in itself) to both positive and negative infinity? I’ve seen this done in complex analysis via the extended complex plane; but I’ve never heard a plausible argument for doing this with only the integers.

P.S. It’s amusing to note that a Java int (and an int in most other programming languages) is not really an integer at all, but rather a member of the group ℤ232. Or at least it’s isomorphic to ℤ232. I’m not sure if there’s a name for the variant of this group in which we only go halfway to the order of the group and then wrap around into negative numbers.

One Response to “Why are the Integers a Cyclic Group?”

  1. John Cowan Says:

    In the early development of mathematical theories, it’s typical to look only at finite sets, and the intuitions (and names) that develop during that stage don’t necessarily apply cleanly to infinite sets. In any finite integral domain, it’s clear that the members of {x | x = 2n} are only about half the domain, but in the integers themselves, the even integers and the integers have exactly the same size. This is counterintuitive but leads to the best overall results. Similarly, treating the integers (and groups isomorphic to them) as cyclic groups gives the best generalizations.

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