Every resource on the web says that a cyclic group "is a group that is generated by a single element". And that's about it, with no concrete example. Lots of technical abstractions and such, but not much in terms of real-world concrete examples, such as using metaphors or common earthly patterns/systems to demonstrate.
The next semi-(non-concrete) example you might encounter is "integer and modular addition". I get the definition of a group. But I don't follow these abstract examples.
Can you show a "hardcoded" example of what a cyclic group is using perhaps some small subset of the integers?
From my reading/parsing of these teachings about cyclic groups, here is what I would say.
- A group might be the even integers, where 2 can be used to generate every other even integer if the group operation is addition. Am I correct here?
- A group might be a square with rotation as the operation. I don't at all get how this can form a cyclic group. I can rotate the square by 1 degree or 10 degrees or 540 degrees (modulo to 180). What does that have to do with cyclic groups? What is the generator? Do I pick a limitation such as "only rotate by 90 degrees", and so 90-degree-rotation is the "generator" which can generate 90, 180, 270 and 360? What if I wanted to rotate by 2 degrees, then 2 is the generator? What if I wanted to allow rotations by 2, 3, 5, and 7? Perhaps it turns out 7 can't give you every rotation possibility, so then 7 would not be a generator? But then 2 wouldn't either because I can't get at the 1-degree angles. Am I on the right track?
Is there always just 1 generator for a cyclic group, or can there be more than one (if I am at all on the right track)?
What is a better, even more concrete example than the ones I tried to provide? Hoping for not something abstract like the definitions/snippets you find on the web, but something with hardcoded values walking me through like I am brand new to math.
When it says "a group generated by a single element", I hear "there is only one unique element in the group (such as with the integers), which can be used to generate all the other elements (such as all the other integers)". In that case, 1 would be the generator, and that's it. I am clearly quite confused.
The most common examples of cyclic groups are $\mathbb{Z}/n\mathbb{Z}$ (under addition) and $\mathbb{Z}$ (likewise). This is because, up to isomorphism, these are all of the cyclic groups.
Correct.
Rotation is not enough for the dihedral groups, but only certain rotations work. The idea is that, after the rotation, the square has to look the exact same as it did before (aside from which vertex/sides are where).
Imagine, for instance, overlapping two drawn squares on each other, and then rotating the top one: after the rotation, it needs to overlap the other one.
For squares, this limits you to $90^\circ$ rotations; for regular $n$-gons in general, you must use $360^\circ/n$ rotations.
The set of rotations here is a cyclic group, of order $n$.
This is somewhat up to your definition of "generator". For instance, one could say that $3$ and $2$ together generate $\mathbb{Z}$ (see: Bezout's identity), but that's not the simplest set of generators.
I think it might be most clear to say "a cyclic group is a group which can be generated by a single element". You may be able to generate the set with multiple elements, and you might even have it where multiple elements on their own are generators ($1$ and $-1$ each individually generate $\mathbb{Z}$), but so long as you can do it with just one generator you're golden.