Applying the abstraction; More Groups

For a very different example, consider the set of integers, the binary operation of addition. (Here the group elements aren't motions, they're just numbers!) We have closure - integers + integer gives an integer. We've got an identity, zero. And every postive number n has an inverse, which is -n, and vice versa. Voila, it's a group! A commutative ⁴ one, too. The fractions under addition and the reals under addition also are groups.

The fractions or the reals, not including zero, under multiplication, are also groups. Here the identify is 1, and inverses are reciprocals. We have to exclude zero, since it has no inverse, and the integers aren't a group - they're closed under multiplication, but you need fractions to get inverses. They'd be another example of a semi-group which wasn't a group, though.

We also find that the even integers under addition are a group. Or the multiples of three, or four, or any integer. So the even integers are a subset of the integers, but a self-contained group under the same group operation. We call this a subgroup, a group which is contained within another. Similarly the positive fractions (or reals) are a subgroup of all the non-zero fractions (or reals) under multiplication. And going back to our earlier examples, we can see that Cyc₃ is a subgroup of Dih₃, as are $\{I,t\}, \{I,l\}, \{Ir\}$. Cyc₆ has the subgroups $\{I,s^3\}, \{I,s^2,s^4\}$.

We can also see that just as a single flip of the triangle expanded the range of positions, from the three rotations to the 6 rotations+reflections, just adding -1 to the group of the positive fractions under multiplication doubles the size of the group, bringing in all the negative fractions. (And likewise for the reals.) Ditto for adding 1 to the even integers under addition.