The structure of groups of finitary transformations
Let V be a vector space over a field k. A linear map g: \(V\to V\) is called finitary, if dim V(g-1)\(<\infty\), or equivalently, if g acts as the identity on a subspace of finite codimension in V. The set of all invertible finitary linear transformations of G is a subgroup FGL(V) of the full group of invertible linear transformations of V. A group is called a k-finitary linear group, if it can be embedded in FGL(V) for some vector space V over k. Examples include (i) any group that can be faithfully represented by matrices of finite size over k (these are here called finite-dimensional k-linear), and (ii) any group of finitary permutations, acting on the associated permutation module over k. A recent and striking theorem of \textit{J. I. Hall} [J. Algebra 119, 337-359 (1988; Zbl 0665.20017)] asserts that if k has characteristic zero and G is an infinite simple locally finite k-finitary linear group, then \(G\cong Alt(\Omega)\) for some set \(\Omega\). The present paper seeks to lay the groundwork for the general study of finitary linear groups, and obtains a lot of information about the locally finite ones, particularly the ``non-modular case'' when the groups concerned have no non-trivial elements of order equal to the characteristic of the underlying field. More general finitary linear groups also receive attention. We now describe the main results in this stimulating paper. Theorem A: Let G be a periodic \((=locally\) finite) k-finitary linear group, and suppose we have the non-modular case (see above). Then (i) G is a subdirect product of irreducible k-finitary linear groups, each of which is an image of G (these are essentially the irreducible constituents of G in a natural sense; note that ``direct product'' means ``restricted direct product'' throughout) (ii) if G is irreducible, then G has a normal subgroup N such that N is a subdirect power of a finite- dimensional k-linear group and G/N is isomorphic to a transitive group of finitary permutations, (iii) if G is irreducible and the corresponding vector space has infinite dimension, then \(G'\) is the unique minimal subnormal irreducible subgroup of G. Statement (ii) of Theorem A should be viewed as two separate statements, since it covers two quite different situations. Let G be irreducible, as in Theorem A(ii), and suppose first that G has a non-trivial reducible normal subgroup H. After some work, it turns out that a form of Clifford's Theorem is available. Let V be the underlying vector space. Then V contains a finite dimensional irreducible H-submodule. The detection of an irreducible H-submodule is not a triviality, as it would be in the finite-dimensional case. It then follows quite easily that V decomposes into homogeneous components under the action of H. These are permuted transitively and finitarily by G, and N can be taken to be the kernel of this permutation action. This case is Proposition 2 of the paper. The more difficult case of Proposition A(ii) is when every non- trivial normal subgroup of G is irreducible. If the corresponding vector space V is infinite-dimensional, it turns out that G is either the alternating group or the full symmetric group on an infinite set \(\Omega\), and V is the commutator submodule of the corresponding permutation module. This is Proposition 3; because of the non-modularity assumption, it follows now that this case can only occur in characteristic zero. Proposition 3 is a substantial generalization of Hall's result referred to above. Its proof involves showing (after some effort) that \(G'\) is simple, then Hall's result can be invoked. Other interesting technical ideas are also involved, for which the reader is referred to the paper.
invertible finitary linear transformations
full group of invertible linear transformations
infinite simple locally finite k-finitary linear group
subdirect product of irreducible k- finitary linear groups
transitive group of finitary permutations
scientific article; zbMATH DE number 4095765
scientific article