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8). 27)). 9) THEOREM. Let A be a simple algebra whose center F is an algebraic number field, and let A be a maximal R-order in A. For a prime ideal P of R, let A p === Mn(D), where D is a skewfield, and set m p == (D: Fp )Y2, the local index of A at P. 2). 37 CLASS GROUPS AND PICARD GROUPS is a cyclic group of order m p Furthermore, m p . = 1 for almost all P. OUTLINE OF PROOF. The group leAp) is an infinite cyclic group generated by rad A p . p If rr denotes a prime element of Rp, then rrAp = (rad Ap)m .

31 n - 2.

Now u(Z) = {in: m = 1, 2, ... , p - I}. For each such m, there is a cyclotomic unit u = (w m - 1)/(w - 1) E u(R), such that u == m (mod P). Thus ii = in u(Z), which proves that u(R) maps onto u(Z). Hence D(A) = 0, as claimed. 6) that Cl A ~ Cl R. This theorem was first proved (in another way) by Rim [33], using results of Reiner. The above proof is due to Milnor. 3) THEOREM. Let V = be a (2, 2)-group. Then D(ZV) = 0. PROOF. Let A = ZV, 1= (t - I)A, J = (t + I)A, Z = Z/2Z. 11) becomes A-Z[s] 1 Z[s] ~ 1 Z[s], where Z[s] is the integral group ring of the cyclic group

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A Characterization of Alternating Groups II by Mazurov V.D.

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