The Amit–Ashurst conjecture for finite metacyclic p-groups

Bera [Line graph characterization of power graphs of finite nilpotent groups, Comm. Algebra 50(11) (2022) 4652–4668] characterized finite nilpotent groups whose power graphs and proper power graphs are line graphs. In this paper, we extend the results of above-mentioned paper to arbitrary finite groups. Also, we correct the corresponding result of the proper power graphs of dihedral groups. Moreover, we classify all the finite groups whose enhanced power graphs are line graphs. We classify all the finite nilpotent groups (except non-abelian [Formula: see text]-groups) whose proper enhanced power graphs are line graphs of some graphs. Finally, we determine all the finite groups whose (proper) power graphs and (proper) enhanced power graphs are the complement of line graphs, respectively.

In this article, it is shown that the normalizer property holds for the following two kinds of finite nilpotent-by-nilpotent groups: (1) G = NwrH is the standard wreath product of N by H, where N is a finite nilpotent group and H is a finite abelian 2-group; (2) G is a finite group having a normal nilpotent subgroup N such that the integral group ring ℤ(G/N) has only trivial units. Our results generalize a result of Yuanlin Li and extend some ones obtained by Juriaans, Miranda, and Robério.

The enhanced power graph [Formula: see text] of a group [Formula: see text] is the graph with vertex set [Formula: see text] such that two vertices [Formula: see text] and [Formula: see text] are adjacent if they are contained in the same cyclic subgroup. We prove that finite groups with isomorphic enhanced power graphs have isomorphic directed power graphs. We show that any isomorphism between undirected power graph of finite groups is an isomorphism between enhanced power graphs of these groups, and we find all finite groups [Formula: see text] for which [Formula: see text] is abelian, all finite groups [Formula: see text] with [Formula: see text] being prime power, and all finite groups [Formula: see text] with [Formula: see text] being square-free. Also, we describe enhanced power graphs of finite abelian groups. Finally, we give a characterization of finite nilpotent groups whose enhanced power graphs are perfect, and we present a sufficient condition for a finite group to have weakly perfect enhanced power graph.

Nilpotent semigroups in the sense of Mal’cev are defined by semigroup identities. Finite nilpotent semigroups constitute a pseudovariety, MN, which has finite rank. The semigroup identities that define nilpotent semigroups lead us to define strongly Mal’cev nilpotent semigroups. Finite strongly Mal’cev nilpotent semigroups constitute a non-finite rank pseudovariety, SMN. The pseudovariety SMN is strictly contained in the pseudovariety MN, but all finite nilpotent groups are in SMN. We show that the pseudovariety MN is the intersection of the pseudovariety BGnil with a pseudovariety defined by a κ-identity. We further compare the pseudovarieties MN and SMN with the Mal’cev product 𝖩ⓜ𝖦nill.

P. Hall's classical equality for the number of conjugacy classes‎ ‎in $p$-groups yields $k(G) ge (3/2) log_2 |G|$ when $G$ is nilpotent‎. ‎Using only Hall's theorem‎, ‎this is the best one can do when $|G| = 2^n$‎. ‎Using a result of G.J‎. ‎Sherman‎, ‎we improve the constant $3/2$ to $5/3$‎, ‎which is best possible across all‎ ‎nilpotent groups and to $15/8$ when $G$ is nilpotent and $|G| ne 8,16$‎. ‎These results are then used to prove that $k(G) > log_3(|G|)$ when $G/N$ is nilpotent‎, ‎under ‎‎‎‎natural conditions on $N trianglelefteq G$‎. ‎Also‎, ‎when $G'$ is nilpotent of class $c$‎, ‎we prove that $k(G) ge (log |G|)^t$ when $|G|$ is large enough‎, ‎depending only on $(c,t)$‎.

Let [Formula: see text] be a finite nilpotent group. We prove the following results. (1) If [Formula: see text] is of class 2 and acts faithfully and irreducibly on an elementary abelian group [Formula: see text], then all nontrivial orbits of [Formula: see text] on [Formula: see text] have sizes larger than [Formula: see text]. (2) If [Formula: see text] is cyclic, then every subgroup of [Formula: see text] intersecting trivially with the center of [Formula: see text] has order less than [Formula: see text]. We also show that a result like (2) cannot be obtained when the hypothesis that [Formula: see text] is cyclic is replaced by the hypothesis that the center of [Formula: see text] is cyclic.

The work in this article was motivated by a question about Fibonacci sequences in groups. Let a, b be arbitrary elements of a group G. We define a Fibonacci sequence in G as for the integers. Let x0 = a, x1 = b and xn+1 = xn−1xn. In [3], Wall investigated the length of the fundamental period of the Fibonacci sequence in a finite cyclic group. Aydin and Smith [1] looked more generally at finite nilpotent groups. The study quickly reduces to finite p-groups. Denote by k(G), the length of the fundamental period of the Fibonacci sequence in G. Aydin and Smith proved the following theorem that generalises a result that Wall obtained in the cyclic case.

The minimal ramification problem may be considered as a quantitative version of the inverse Galois problem. For a nontrivial finite group $G$, let $m(G)$ be the minimal integer $m$ for which there exists a $G$-Galois extension $N/\mathbb{Q}$ that is ramified at exactly $m$ primes (including the infinite one). So, the problem is to compute or to bound $m(G)$.In this paper, we bound the ramification of extensions $N/\mathbb{Q}$ obtained as a specialization of a branched covering $\unicode[STIX]{x1D719}:C\rightarrow \mathbb{P}_{\mathbb{Q}}^{1}$. This leads to novel upper bounds on $m(G)$, for finite groups $G$ that are realizable as the Galois group of a branched covering. Some instances of our general results are: $$\begin{eqnarray}1\leqslant m(S_{k})\leqslant 4\quad \text{and}\quad n\leqslant m(S_{k}^{n})\leqslant n+4,\end{eqnarray}$$ for all $n,k>0$. Here $S_{k}$ denotes the symmetric group on $k$ letters, and $S_{k}^{n}$ is the direct product of $n$ copies of $S_{k}$. We also get the correct asymptotic of $m(G^{n})$, as $n\rightarrow \infty$ for a certain class of groups $G$.Our methods are based on sieve theory results, in particular on the Green–Tao–Ziegler theorem on prime values of linear forms in two variables, on the theory of specialization in arithmetic geometry, and on finite group theory.

Thin bases for finite nilpotent groups

Suppose that $A,B$ are non-empty subsets of the finite nilpotent group $G$. If $A\not=B$, then the cardinality of the restricted sumset $$A\mathbin\dotplus B=\{a+b:a\in A,\, b\in B,\, a\neq b\} $$ is at least $$\min\{p(G),|A|+|B|-2\},$$ where $p(G)$

The probability distribution of word maps on finite groups

Finite non-cyclic nilpotent group whose number of subgroups is minimal

Finite nilpotent groups having exactly four conjugacy classes of non-normal subgroups are classified.

On the Number of Generators and Composition Length of Finite Linear Groups

Finite group theorists have been interested in counting groups of prime-power order, as a preliminary step to counting groups of any finite order and to assist in explicitly listing such groups. In 1960, G. Higman considered when the functions giving the number of groups of prime-power order pn, for fixed n and varying p, is of a particular form, called polynomial on residue classes (PORC). The suggestion that such counting functions are PORC is known as Higman's PORC conjecture. In his 1960 paper [4] he proved that a certain class of groups of prime-power order, now called exponent-p class two groups, have counting functions that are PORC, but did not furnish explicit PORC functions.