Automorphism groups with some finiteness conditions

Year, pages: 2017, 831 - 852

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 $\Aut (A)$.

Pióro, K. 2017. Automorphism groups with some finiteness conditions. In Mathematica Slovaca, vol. 67, no.4, pp. 831-852. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0014

Pióro, K. (2017). Automorphism groups with some finiteness conditions. Mathematica Slovaca, 67(4), 831-852. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0014