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Groups with no nontrivial linear representations. (English) Zbl 0815.20026
The author discusses groups \(G\) for which there are no nontrivial homomorphisms of \(G\) into \(\text{GL}(n,k)\) for any positive integer \(n\), either for a fixed field \(k\) (and he calls such groups \(G\) counter-\(k\)- linear) or for any field \(k\) of fixed characteristic \(p \geq 0\) (the counter-\(p\)-linear groups), or for any field \(k\) (the counter-linear groups).
The paper starts by reviewing the examples of such groups already in the literature, and there are a substantial number. The author then proves a number of criteria for counter-linearity. For example if \(G\) is a group with no non-cyclic free subgroups then \(G\) is counter-0-linear if and only if \(G\) is perfect and there exists some field \(k\) of characteristic 0 with \(G\) counter-k-linear. A locally soluble group is counter-linear if and only if it is perfect. Finally he proves that many acyclic groups are counter-linear. For example the binate groups are counter-0-linear and \(\text{Sym}(X)\) and \(\text{GL}(V)\) are counter-linear for any infinite set \(X\) and infinite-dimensional vector space \(V\).
Unfortunately, this reviewer is unconvinced by some of the arguments and a couple of results. At least (2.2 and A.1ii & iv) need some modification to their statements.

MSC:
20F29 Representations of groups as automorphism groups of algebraic systems
20C32 Representations of infinite symmetric groups
20E36 Automorphisms of infinite groups
20F38 Other groups related to topology or analysis
20C15 Ordinary representations and characters
20E34 General structure theorems for groups
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