1, and let ø denote Let m and n be positive integers with gcd(m, n) Euler's o-function (Definition 1.45). Consider the group Z/nZ × Z/mZ. (a) Show that (a,b) is a generator for the group Z/nZ x Z/mZ if and only if a and b are, respectively, generators for Z/nZ and Z/mZ. (b) Show that the number of generators of the group Z/nZ × Z/mZ is $(n)o(m). (c) Prove that for relatively prime integers m and n, $(nm) = ¢(n)ø(m).

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A:

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A:

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A:

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Q: 9*. Use Lagrange's theorem to prove: If g E G, G a finite group, then gG| = 1.

A: To prove the given statement, assume the following.

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