We study the higher Frobenius-Schur indicators of the representations of the Drinfel'd double of a finite group G, in particular the question as to when all the indicators are integers. This turns out to be an interesting group-theoretic question. We show that many groups have this property, such as alternating and symmetric groups, P SL 2 (q), M 11 , M 12 and regular nilpotent groups. However we show there is an irregular nilpotent group of order 5 6 with non-integer indicators.

We study maximal associative subalgebras of an arbitrary finite-dimensional associative algebra B over a field {\mathbb{K}} and obtain full classification/description results of such algebras. This is done by first obtaining a complete classification in the semisimple case and then lifting to non-semisimple algebras. The results are sharpest in the case of algebraically closed fields and take special forms for algebras presented by quivers with relations. We also relate representation theoretic properties of the algebra and its maximal and other subalgebras and provide a series of embeddings between quivers, incidence algebras and other structures which relate indecomposable representations of algebras and some subalgebras via induction/restriction functors. Some results in literature are also re-derived as a particular case, and other applications are given.

We investigate left and right co-Frobenius coalgebras and give equivalent characterizations which prove statements dual to the characterizations of Frobenius algebras. We prove that a coalgebra is left and right coFrobenius if and only if C ∼ = Rat(C * C * ) as right C * -modules and also that this is equivalent to the fact that the functors Hom K (−, K) and Hom C * (−, C * ) from M C to C * M are isomorphic. This allows a definition of a left-right symmetric concept of co-Frobenius coalgebras that is perfectly dual to the one of Frobenius algebras and coincides to the existing notion left and right co-Frobenius coalgebra.