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This leads us to the notion of an isomorphism:23 a bijection from one group to another which respects the group structures. More precisely, a bijection f : G → H is an isomorphism if f ( g1 ∗ g2 ) = f ( g1 )◦ f ( g2 ) for all g1 , g2 ∈ G. Another tool for understanding the internal structure of a group is the order24 | g| of an element g: the smallest natural number n such that gn = e (or ng = e if we’re using additive notation). If no such n exists, we say the element is of infinite order. 27 Some groups can be reconstructed from a proper subset of elements of the group.

That is, this group completely describes the symmetry of an equilateral triangle. This group is the dihedral group, and there are two common conventions for what to call it. 6: Axes of symmetry of an symmetry of a regular 3–sided polygon. In this book, we’ll adopt the equilateral triangle groups latter convention, but you should be aware that some books follow the other one. 5. · e r r2 e e r r2 r r2 e r r r2 r2 e m1 m1 m2 m3 m2 m2 m3 m1 m3 m3 m1 m2 m1 m1 m3 m2 e r2 r m2 m2 m1 m3 r e r2 21 m3 m3 m2 m1 r2 r e More generally, we can do the same thing with a regular n–gon.

That is, li |n. Repeating this argument for all the cycles σ1 , . . , σk we see that if π n = ι then li |n for 1 i k. Conversely, suppose that n is such that li |n for 1 π n = σin . . σkn = ι . . ι = ι. i k. Then Therefore, π n = ι if and only if li |n for 1 i k. So in particular, n must be the smallest positive integer which divides each of the li , which is to say that it must be the lowest common multiple lcm(li , . . , lk ). So |σ | = l (σ ) = 3, but |τ | = lcm(2, 3) = 6. Now let’s look again at the permutation σ = (2 3 5).

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