Related papers: Groups with at most twelve subgroups

We prove a general divisibility theorem that implies, e.g., that, in any group, the number of generating pairs (as well as triples, etc.) is a multiple of the order of the commutator subgroup. Another corollary says that, in any associative…

We make a list of finite simple groups whose group rings over a given field are serial.

We introduce the notion of numerical semigroups generated by concatenation of arithmetic sequences and show that this class of numerical semigroups exhibit multiple interesting behaviours.

We are interested in formulas for the number of elements in certain classes of numerical semigroups

In this paper we study grouplike monoids, these are monoids that contain a group to which we add an ordered set of idempotents. We classify finite categories with two objects having grouplike endomorphism monoids, and we give a count of…

We present exact and heuristic algorithms that find, for a given family of graphs, a graph that contains each member of the family as an induced subgraph. For $0 \leq k \leq 6$, we give the minimum number of vertices $f(k)$ in a graph…

A set of proper subgroups is a covering for a group if its union is the whole group. The minimal number of subgroups needed to cover $G$ is called the covering number of $G$, denoted by $\sigma(G)$. Determining $\sigma(G)$ is an open…

We prove an upper bound for the number of cyclic transitive subgroups in a finite permutation group and clarify the structure of the groups for which this bound becomes sharp. We also give an application in the theory of number fields.

For the first time we represent every finite group in the form of a graph in this book. The authors choose to call these graphs as identity graph, since the main role in obtaining the graph is played by the identity element of the group.…

We deduce a simple representation and the invariant factor decompositions of the subgroups of the group $\Bbb{Z}_m \times \Bbb{Z}_n$, where $m$ and $n$ are arbitrary positive integers. We obtain formulas for the total number of subgroups…

In this paper, we provide some conditions of (super)-solvability and nilpotency of a finite group $G$ based on its number of subgroups $Sub(G)$. Our results generalize the classification of finite groups with less than $20$ subgroups by…

We deduce direct formulas for the total number of subgroups and the number of subgroups of a given order of the group $\Bbb{Z}_m\times \Bbb{Z}_n \times \Bbb{Z}_r \times \Bbb{Z}_s$, where $m,n,r,s\in \Bbb{N}$. The proofs are by some simple…

We introduce the notion of the depth of a finite group $G$, defined as the minimal length of an unrefinable chain of subgroups from $G$ to the trivial subgroup. In this paper we investigate the depth of (non-abelian) finite simple groups.…

A nut graph is a simple graph of order 2 or more for which the adjacency matrix has a single zero eigenvalue such that all non-zero kernel eigenvectors have no zero entry (i.e. are full). It is shown by construction that every finite group…

For a finite group $G$ let $\sigma(G)$ (the "sum" of $G$) be the least number of proper subgroups of $G$ whose set-theoretical union is equal to $G$, and $\sigma(G)=\infty$ if $G$ is cyclic. We say that a group $G$ is $\sigma$-elementary if…

In this paper, we provide new criteria for the solvability and supersolvability of a finite group based on its number of cyclic subgroups. A finite group G is called n-cyclic if it contains n cyclic subgroups. This paper also partially…

Let $\lambda(G)$ be the maximum number of subgroups in an irredundant covering of a finite group $G$. We prove that the finite groups with $\lambda(G)=|G|-t$, where $t\leq 5$, are solvable, and classify such groups.

We describe general methods for enumerating subsemigroups of finite semigroups and techniques to improve the algorithmic efficiency of the calculations. As a particular application we use our algorithms to enumerate all transformation…

We show that every finite group $G$ of size at least $3$ has a nilpotent subgroup of class at most $2$ and size at least $|G|^{1/32\log\log|G|}$. This answers a question of Pyber, and is essentially best possible.

We examine $p$-groups with the property that every non-normal subgroup has a normalizer which is a maximal subgroup. In particular we show that for such a $p$-group $G$, when $p=2$, the center of $G$ has index at most 16 and when $p$ is odd…