Abstract
Let D be a division ring with infinite center F. By a well known result of Amitsur, if satisfies a group identity, then D is commutative. Now assume that D has an involution * of the first kind. In this paper, among other results, we show that if satisfies a *-group identity, then either D is commutative or dim F D = 4 and * is of the symplectic type. As a result, let N be a *-invariant normal subgroup of such that all symmetric elements of N are central (this is the case when, for example, each symmetric element of N is bounded Engel). Then either N is central or and * is of the symplectic type.
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