Section 1.4. Lagrange’s Theorem 37 Example 1.55. It is easy to see that the square of each element in the group U(I8) = [1], [3], [5], [7] is [1] [thus, U(I8) resembles the four-group V], while U(I10) = [1], [3], [7], [9] is a cyclic group of order 4 with generator [3] [were the term isomorphism available (we introduce it in the next section), we would say that U(I8) is isomorphic to V and U(I10) is isomorphic to I4]. See Example 1.60. Theorem 1.56 (Wilson’s Theorem). An integer p is prime if and only if (p − 1)! ≡ −1 mod p. Proof. Assume that p is prime. If a1,a2,...,an is a list of all the elements of a finite abelian group G, then the product a1a2 · · · an is the same as the product of all elements a with a2 = 1, for any other element cancels against its inverse. Since p is prime, Ip × has only one element of order 2, namely, [−1] (if p is prime and x2 ≡ 1 mod p, then x = [±1]). It follows that the product of all the elements in Ip ×, namely, [(p − 1)!], is equal to [−1] therefore, (p − 1)! ≡ −1 mod p. Conversely, assume that m is composite: there are integers a and b with m = ab and 1 a ≤ b m. If a b, then m = ab is a divisor of (m−1)!, and so (m−1)! ≡ 0 mod m. If a = b, then m = a2. If a = 2, then (a2 − 1)! = 3! = 6 ≡ 2 mod 4 and, of course, 2 ≡ −1 mod 4. If 2 a, then 2a a2, and so a and 2a are factors of (a2 − 1)! therefore, (a2 − 1)! ≡ 0 mod a2. Thus, (a2 − 1)! ≡ −1 mod a2, and the proof is complete. • Remark. We can generalize Wilson’s Theorem in the same way that Euler’s Theo- rem generalizes Fermat’s Theorem: replace U(Ip) by U(Im). For example, if m ≥ 3, we can prove that U(I2m ) has exactly 3 elements of order 2, namely, [−1], [1+2m−1], and [−(1 + 2m−1)] (Rotman, An Introduction to the Theory of Groups, p. 121). It follows that the product of all the odd numbers r, where 1 ≤ r 2m, is congruent to 1 mod 2m, because (−1)(1 + 2m−1)(−1 − 2m−1) = (1 + 2m−1)2 = 1 + 2m + 22m−2 ≡ 1 mod 2m. Exercises ∗ 1.38. Let H be a subgroup of a group G. (i) Prove that right cosets Ha and Hb are equal if and only if ab−1 ∈ H. (ii) Prove that the relation a ≡ b if ab−1 ∈ H is an equivalence relation on G whose equivalence classes are the right cosets of H. 1.39. (i) Define the special linear group by SL(2, R) = {A ∈ GL(2, R) : det(A) = 1}. Prove that SL(2, R) is a subgroup of GL(2, R). (ii) Prove that GL(2, Q) is a subgroup of GL(2, R).

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