Saturday, 10 January 2015

abstract algebra - (Infinite) Non-abelian group with only linear characters



If G is an abelian group, then every irreducible character has dimension one (i.e. is linear), for finite group we also have a converse. Do we have a converse for infinite groups? Or:




Does there exists an infinite group all whose irreducible characters are linear, which is not abelian?




I append a proof that G abelian implies all irreducible characters are linear: If G is an abelian group and V an irreduble C[G]-module, then by Schur's lemma we have dimHomC[G](V,V)=1. As G is abelian, every g corresponds to a C[G]-homomorphism. Hence in its action on V could be identified with some homomorphism of the form λIn with n=dimV. If n>1 then such a map would leave every space invariant, also the ones of dimension <n, hence n=1.



Answer



Yes. Take an infinite simple group G, such as PSL3(K) for a sufficiently large field K of sufficiently large characteristic, with cardinality strictly larger than C. Then every finite-dimensional representation of G over C is trivial, so its only irreducible character is the trivial one.



There are more complicated examples of finitely generated or even finitely presented groups all of whose finite-dimensional representations are trivial.


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