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Mathematica Slovaca

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Volume 67, Issue 4

Issues

Automorphism groups with some finiteness conditions

Konrad Pióro
Published Online: 2017-07-14 | DOI: https://doi.org/10.1515/ms-2017-0014

Abstract

All considered groups are torsion or do not contain infinitely generated subgroups. If such a group G acts transitively on a connected algebra A, then all elements of A have the same stabilizer, so this stabilizer is a normal subgroup (it is also shown that these facts are not true for arbitrary groups). Hence the automorphism group Aut(A) is a homomorphic image of G. In particular, if the action of G is, in addition, faithful, then G is isomorphic to Aut(A). By these results we first obtain that for each (not necessarily connected) algebra A, if all of its Aut(A)-orbits are connected, then Aut(A) is a subdirect product of automorphism groups of these orbits. Secondly, connected components of Aut(A)-orbits have automorphism extension property.

Next, we show that if A is a connected algebra such that Aut(A) acts transitively on it, then the group structure of Aut(A) may be transported on A such that the left multiplications are all automorphisms of A and the right multiplications are all unary term operations of A. Hence all the unary term operations of A are bijective, so they generate a subgroup of the group of all bijections of the carrier of A. It is shown that this group is anti-isomorphic to Aut(A). Thus the Birkhoff’s construction of an algebra with a given group of automorphisms is unique in some sense when restricted to groups we consider here.

The Birkhoff’s construction can be slightly modified so as to obtain a smaller set of operations. In fact, it is enough to take the right multiplications by generators. Moreover, we show that this is the best possible lower bound for the number of unary operations in the case of groups considered here. If we admit non-unary operations, then for finite and countable groups we can reduce the number of operations to one binary operation. On the other hand, if A is a connected algebra such that Aut(A) is torsion and acts on A transitively, then each element generates A. Hence if A is such an algebra with an uncountable group Aut(A), then the cardinality of the set of operations of A is greater or equal than the cardinality of Aut(A).

MSC 2010: Primary 08A35; Secondary 08A55; 20B25; 20B27

Keywords: action of group; automorphism of universal algebra; automorphism group; connected algebra; partial algebra; partial subalgebras; torsion group; finitely generated group; representation of group

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About the article


Received: 2014-11-24

Accepted: 2015-10-08

Published Online: 2017-07-14

Published in Print: 2017-08-28


Citation Information: Mathematica Slovaca, Volume 67, Issue 4, Pages 831–852, ISSN (Online) 1337-2211, ISSN (Print) 0139-9918, DOI: https://doi.org/10.1515/ms-2017-0014.

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