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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).

##### MSC:
 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
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