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# Isomorphic groups

Let . Define the group

Prove that where is the dihedral group.

Solution

Using  or equivalently we can write each element of  in the form where . Using we may assume that . Using we may also assume that . It is easy to prove inductively that .

Let . We prove that . Obviously . Furthermore,  since

If (such that ) then hence and therefore or . If then hence which is a contradiction since .

Therefore,

which is precisely the dihedral group with elements.