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# Question on U groups and primes?

Let p and q be distinct odd primes. Prove that U(pq) has exactly 3 elements of order 2.

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- rodolfo riverolLv 61 decade agoFavorite Answer
A theorem of Gauss is that U(p) = Z/(p)* is a multiplicative cyclic group of order p-1. If g is a generator then g^((p-1)/2) = -1 is the only order 2 element.

U(pq) = Z/(pq)* ~ Z/(p)* x Z/(q)* by the chinese remainder theorem. So (-1,1), (1, -1) and (-1,-1) are the only elements of order 2.

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