4.1.7. Let G = Dg act on N = D8 by conjugation. Make an 8 × 8 table, where the rows are indexed by the elements of G and the columns are indexed 2. The Cayley Graph of a Group Action* 91 by the elements of . The entry in row g and column x is the result of the action of g on x (that is g. x.) Complete the table. (a) In each row do you get every element of N? Does any element of N ever appear in any row more than once? Is this a coincidence? (b) Does an element of Never appear in any column more than once? Is it possible for a column to contain only one element of ? When does this happen? (c) Can two rows be identical? (d) Is it possible to tell the order of the elements of the group from this table?

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