On loops whose inner permutations commute.

*(English)*Zbl 1101.20034Let \(Q\) be a loop, \(G=M(Q)\) its multiplication group, and \(H=I(Q)\) its inner mapping group. The paper studies certain aspects of \(G\) under the assumption that \(H\) is Abelian.

Denote by \(K\) the normal closure of \(H\) in \(G\). Then \(K=G'H\), for every loop (Proposition 3.1). If \(H\) is Abelian, then \(Z(G)\cap K\neq 1\), \(Z(K)\neq 1\) and \(Z(K)H\neq H\) (Proposition 3.6). In Section 4 one assumes, in addition, the existence of \(P\leq Z(G)\cap K\) with \(PH\trianglelefteq K\). This gives a number of consequences; for example that \(G'''=1\) and that \(K\) is nilpotent of class at most two. In the last section one proves, amongst others, that if \(Q\) is nilpotent of class at least three, then every prime dividing \(|H|\) divides \(|Q|\).

The paper uses the language of \(H\)-connected transversals. That makes it look very formal. However, when a translation is made into the language of structural loop theory, the statements usually acquire a clear meaning. In a few cases one even discovers that rather obvious facts are being proved – for example Lemma 2.10 is a veiled form of saying that a loop is a group if and only if the left and the right translations commute.

Denote by \(K\) the normal closure of \(H\) in \(G\). Then \(K=G'H\), for every loop (Proposition 3.1). If \(H\) is Abelian, then \(Z(G)\cap K\neq 1\), \(Z(K)\neq 1\) and \(Z(K)H\neq H\) (Proposition 3.6). In Section 4 one assumes, in addition, the existence of \(P\leq Z(G)\cap K\) with \(PH\trianglelefteq K\). This gives a number of consequences; for example that \(G'''=1\) and that \(K\) is nilpotent of class at most two. In the last section one proves, amongst others, that if \(Q\) is nilpotent of class at least three, then every prime dividing \(|H|\) divides \(|Q|\).

The paper uses the language of \(H\)-connected transversals. That makes it look very formal. However, when a translation is made into the language of structural loop theory, the statements usually acquire a clear meaning. In a few cases one even discovers that rather obvious facts are being proved – for example Lemma 2.10 is a veiled form of saying that a loop is a group if and only if the left and the right translations commute.

Reviewer: Aleš Drápal (Praha)