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On finite loops whose inner mapping groups are Abelian. (English) Zbl 1012.20068
Given a loop \((Q,\cdot)\), for any \(a\in Q\) let \(L_a\) and \(R_a\) be the left and the right translations by \(a\) and let \(M(Q):=\langle\{L_a,R_a\mid a\in Q\}\rangle\) be the multiplication group of \((Q,\cdot)\). If we denote by \(I(Q):=\{\gamma\in M(Q)\mid\gamma(e)=e\}\) (where \(e\) is the neutral element of the loop), then \(I(Q)\) is the so-called inner mapping group of the loop \((Q,\cdot)\) (if \((Q,\cdot)\) is a group \(I(Q)\) coincides with the inner automorphism group of \(Q\)). In this paper the author investigates the structure of \(I(Q)\) in particular, he addresses the problem of finding classes of finite Abelian groups possibly isomorphic to \(I(Q)\), generalizing the analogous problem for groups which has been completely solved by Baer (see the reference quoted in the note).
The author reaches the following results: For a finite loop \((Q,\cdot)\): 1. \(I(Q)\) is never isomorphic to the direct product \(C_{p^k}\times C_p\), where \(p\) is an odd prime number and \(k\geq 2\) (\(C_n\) denotes the cyclic group of order \(n\)). 2. \(I(Q)\) is never isomorphic to \((C_{p^k}\times C_p)\times D\) where \(D\) is an Abelian \(q\)-group and \(p\) and \(q\) are two prime numbers such that \(p\) is odd and \(q\) does not divide \(|Q|\), and \(k\geq 2\).
These results are obtained by resorting to general group theoretical techniques via a crucial link provided by a theorem which allows a group \(G\) to be isomorphic to the multiplication group of a loop if and only if there exists a subgroup \(H\) satisfying some particular conditions (see Theorem 2.1).

20N05 Loops, quasigroups
20K01 Finite abelian groups
20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
20F29 Representations of groups as automorphism groups of algebraic systems
Full Text: DOI
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