ON SANOV 4TH-COMPOUNDS OF A GROUP

摘要

In his elegant inductive proof that every finitely generated group of exponent 4 is finite, Sanov used the following construction. Let M be a group and let u be an involution in M. We form a group S_u(M,a) by means of the relations a~2 = u and (ma)~4 = 1 for every m ∈ M. When u = 1, we write S_0(M,a) for the corresponding group. We call S_u(M,a) a Sanov compound and there is one for every conjugacy class of involutions in M. Sanov proved that for finite M of order m, every Sanov compound S_u(M,a) has finite order at most m~(m+1). (See, for example, [2, Theorem 18.3.1] or [3, Theorem 14.2.4].) Here we establish some general results concerning S_u(M,a). For example, if M is infinite cyclic, then S_0 (M,a) is the extension of a countable elementary abelian 2-group by the infinite dihedral group. If M is cylic of order 3, then S_0(M,a) is isomorphic to 54. For M = A_4, S_0(M,a) has order 2~9 · 3, while S_u(M,a) has order 2~6 · 3 for u = (1,2) (3,4).