In this window, all groups are assumed finite. Here we collect a number of results that play a significant role in the book (further material of an elementary nature that we sometimes take for granted is easily available in textbooks such as [H], [R] and [A]).

Finite Group Theory

Structure of a group and the structure of its lattice of subgroups

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Microsoft Visual Studio 在电阻箱检定结果处理中的应用

Let χ be a Dirichlet character (mod n) with conductor d. In a quite recent paper Zhao and Cao deduced the identity ∑k=1n(k − 1,n)χ(k) = φ(n)τ(n/d), which reduces to Menon’s identity if χ is the principal character (mod n). We generalize the above identity by considering even functions (mod n), and offer an alternative approach to prove. We also obtain certain related formulas concerning Ramanujan sums.

Menon-type identities concerning Dirichlet characters

If G is a finite group, then ψ(G) denotes the sum of orders of all elements of G and if k is a positive integer, then Ck denotes a cyclic group of order k. Moreover, ψ(Ck) will be sometimes denoted...

Sums of element orders in groups of order 2m with m odd

https://www.math.uaic.ro/~martar/pdf/articole/articol%2027.pdf

On the number of fuzzy subgroups of finite abelian groups

Subgroup commutativity degrees of finite groups

In this article, we introduce and study two classes of finite groups for which the orders of their subgroups satisfy a certain inequality. These are closely connected to some well-known arithmetic classes of natural numbers.

/pdf/finite-groups-determined-by-an-inequality-of-the-orders-of-4gu3f4gm5v.pdf

Finite groups determined by an inequality of the orders of their subgroups

In this note we give a generalization of the well-known Menon's identity. This is based on applying the Burnside's lemma to a certain group action.

A generalization of Menon's identity

In this short note we describe the finite groups G having |G| − 1 cyclic subgroups. This leads to a nice characterization of the symmetric group S3.

https://www.math.uaic.ro/~martar/pdf/articole/articol%2065.pdf

Marius Tărnăuceanu

Papers

On the number of fuzzy subgroups of finite abelian groups

Subgroup commutativity degrees of finite groups

Finite groups determined by an inequality of the orders of their subgroups

A generalization of Menon's identity

Finite Groups With a Certain Number of Cyclic Subgroups