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Minimal formations of universal algebras

Wenbin Guo, K.P. Shum (2001)

Discussiones Mathematicae - General Algebra and Applications

A class ℱ of universal algebras is called a formation if the following conditions are satisfied: 1) Any homomorphic image of A ∈ ℱ is in ℱ; 2) If α₁, α₂ are congruences on A and A / α i ∈ ℱ , i = 1,2, then A/(α₁∩α₂) ∈ ℱ. We prove that any formation generated by a simple algebra with permutable congruences is minimal, and hence any formation containing a simple algebra, with permutable congruences, contains a minimum subformation. This result gives a partial answer to an open problem of Shemetkov and Skiba...

Multiplicative Dedekind η -function and representations of finite groups

Galina Valentinovna Voskresenskaya (2005)

Journal de Théorie des Nombres de Bordeaux

In this article we study the problem of finding such finite groups that the modular forms associated with all elements of these groups by means of a certain faithful representation belong to a special class of modular forms (so-called multiplicative η - products). This problem is open.We find metacyclic groups with such property and describe the Sylow p -subgroups, p ≠ 2 , for such groups. We also give a review of the results about the connection between multiplicative η -products and elements of finite orders...

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