In analogy with Examples 8.7 and 8.10, consider a regular plane n-gon for n ≥ 3. Each way that two copies of such an n-gon can be placed, with one covering the other, corresponds to a certain permutation of the vertices. The set of these permutations in a group, the nth dihedral group D_n• under permutation multiplication. Find the order of this group D_n. Argue geometrically that this group has a subgroup having just half as many elements as the whole group has.

Data from in example 8.7 

An interesting example for us is the group S3 of 3! = 6 elements. Let the set A be { 1, 2, 3}. We list the permutations of A and assign to each a subscripted Greek letter for a name. The reasons for the choice of names will be clear later. Let

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2 PO = (1 ₂²3). Po 2 P1 = 1 2 3 2 3 1 2 3 - (²) 3 1 2 P2 = (1 = - (1 32). H2 = из 1 2 3 2 1 2 3 1, 2 3 1 3