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On normal subgroups of \(D^*\) whose elements are periodic modulo the center of \(D^*\) of bounded order. (English) Zbl 1325.16033

Let \(D\) be a division ring with center \(F\) and \(N\) a normal subgroup of the multiplicative group \(D^*\) of \(D\). In the article under review, the author proves that if there exists a positive integer \(d\) such that \(x^d\in F\) for any \(x\in N\), then \(N\) is contained in \(F\), which answers partially a conjecture proposed by I. N. Herstein [Isr. J. Math. 31, 180-188 (1978; Zbl 0394.16015)]. The idea of the proof is to show that \(N\) satisfies a nontrivial generalized rational identity over \(D\).

MSC:

16U60 Units, groups of units (associative rings and algebras)
16K40 Infinite-dimensional and general division rings
16K20 Finite-dimensional division rings
16R50 Other kinds of identities (generalized polynomial, rational, involution)
20E15 Chains and lattices of subgroups, subnormal subgroups
20E07 Subgroup theorems; subgroup growth

Citations:

Zbl 0394.16015
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