A note on the Chermak-Delgado lattice of a finite group
In this note we describe the structure of finite groups G whose Chermak-Delgado lattice is the interval [G/Z(G)] = {H \in L(G) \mid Z(G)\leq H\leq G}.
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arXiv subjects
Marius Tărnăuceanu
Publications and source records attributed to Marius Tărnăuceanu.
On some probabilistic aspects of (generalized) dicyclic groups
In this paper we study probabilistic aspects such as subgroup commutativity degree and cyclic subgroup commutativity degree of the (generalized) dicyclic groups. We find explicit formulas for these concepts and we provide another example of a class of groups whose (cyclic) subgroup commutativity degree vanishes asymptotically.
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A characterization of elementary abelian 2-groups
In this note we give a characterization of elementary abelian 2-groups in terms of their maximal sum-free subsets.
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The Chermak-Delgado lattice of ZM-groups
In this note we prove that the Chermak-Delgado lattice of a ZM-group is a chain of length $0$. A similar conclusion is obtained for all dihedral groups $D_{2m}$ with $m\neq 4$.
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Finite groups with a certain number of cyclic subgroups II
In this note we describe the finite groups $G$ having $|G|-2$ cyclic subgroups. This partially solves the open problem in the end of \cite{3}.
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Factorization numbers of finite rank $3$ abelian $p$-groups
In this short note we give a formula for the factorization number $F_2(G)$ of a finite rank $3$ abelian $p$-group $G$. This extends a result in our previous work \cite{9}.
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A note on a metric associated to certain finite groups
In this short note we introduce a new metric on certain finite groups. It leads to a class of groups for which the element orders satisfy an interesting inequality. This extends the class CP_2 studied in our previous paper [16].
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The number of chains of subgroups of a finite elementary abelian $p$-group
In this short note we give a formula for the number of chains of subgroups of a finite elementary abelian $p$-group. This completes our previous work [5].
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A new equivalence relation to classify the fuzzy subgroups of finite groups
In this paper a new equivalence relation $\approx$ to classify the fuzzy subgroups of finite groups is introduced and studied. This generalizes the equivalence relation $\sim$ defined on the lattice of fuzzy subgroups of a finite group that has been used in our previous papers (see e.g. \cite{16,24}). Explicit formulas for the number of distinct fuzzy subgroups with respect to $\approx$ are obtained in some particular cases.
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Cyclicity degrees of finite groups
We introduce and study the concept of cyclicity degree of a finite group $G$. This quantity measures the probability of a random subgroup of $G$ to be cyclic. Explicit formulas are obtained for some particular classes of finite groups. An asymptotic formula and several minimality/maximality results on cyclicity degrees are also inferred.
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An inequality detecting nilpotency of finite groups
In this short note, we provide an inequality that holds in any finite group, only involving the orders of the elements; we prove that equality holds if and only if the group is nilpotent.
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