Exercises

**1.** [*02*] Consider the transformation of {0*,* 1*,* 2*,* 3*,* 4*,* 5*,* 6} that replaces *x* by 2*x* mod 7. Show that this transformation is a permutation, and write it in cycle form.

**2.** [*10*] The text shows how we might set (*a, b, c, d, e, f*) ← (*c, d, f, b, e, a*) by using a series of replacement operations (*x* ← *y*) and one auxiliary variable *t*. Show how to do the job by using a series of *exchange* operations (*x ↔ y*) and no auxiliary variables.

**3.** [*03*] Compute the product , and express the answer in two-line notation. (Compare with Eq. (4).)

**4.** [*10*] Express (*a b d*)(*e f*) (*a c f*) (*b d*) as a product of disjoint cycles.

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