Related papers: On the probability distribution associated to comm…

For $G$ a finite group, let $d_2(G)$ denote the proportion of triples $(x, y, z) \in G^3$ such that $[x, y, z] = 1$. We determine the structure of finite groups $G$ such that $d_2(G)$ is bounded away from zero: if $d_2(G) \geq \epsilon >…

Word maps provide a wealth of information about finite groups. We examine the connection between the probability distribution induced by a word map and the underlying structure of a finite group. We show that a finite group is nilpotent if…

We enumerate the fibres of commutator word maps over p-groups of nilpotency class less than p with exponent p. We also give some examples and enumerate the fibre sizes of all word maps over p-groups of class 2 with exponent p.

The word $w=[x_{i_1},x_{i_2},\dots,x_{i_k}]$ is a simple commutator word if $k\geq 2, i_1\neq i_2$ and $i_j\in \{1,\dots,m\}$, for some $m>1$. For a finite group $G$, we prove that if $i_{1} \neq i_j$ for every $j\neq 1$, then the verbal…

Let G be a finite group of nilpotency class 2 and w a group word. In this short paper we show that the probability that a random n-tuple of elements from G satisfies w is at least one over the order of G. This answers a special case of a…

We study the probability of a given element, in the commutator subgroup of a group, to be equal to a commutator of two randomly chosen group elements, and compute explicit formulas for calculating this probability for some interesting…

Given a group $G$, we write $g^G$ for the conjugacy class of $G$ containing the element $g$. A theorem of B. H. Neumann states that if $G$ is a group in which all conjugacy classes are finite with bounded size, then the commutator subgroup…

Given a finite group $G$, we denote by $\nu(G)$ the probability that two randomly chosen elements of $G$ generate a nilpotent subgroup. We prove that if $\nu(G)>1/12,$ then $G$ is solvable.

It is proved that, for a prime $p>2$ and integer $n\geq 1$, finite $p$-groups of nilpotency class $3$ and having only two conjugacy class sizes $1$ and $p^n$ exist if and only if $n$ is even; moreover, for a given even positive integer,…

We classify finite $p$-groups, upto isoclinism, which have only two conjugacy class sizes $1$ and $p^3$. It turns out that the nilpotency class of such groups is $2$.

Let $N(G)$ be the set of conjugacy classes sizes of $G$. We prove that if $N(G)=\Omega\times \{1,n\}$ for specific set $\Omega$ of integers, then $G\simeq A\times B$ where $N(A)=\Omega$, $N(B)=\{1,n\}$, and $n$ is a power of prime.

For a finite group $G$ and a non-negative integer $d$, denote by $\Omega_d(G)$ the number of functions $G^d\rightarrow G$ that are induced by substitution into a word with variables among $X_1,\ldots,X_d$. In this note, we show that…

Let $G$ be a finite group and $M(G)$ be the subgroup of $G$ generated by all non-central elements of $G$ that lie in the conjugacy classes of the smallest size. Recently several results have been proved regarding the nilpotency class of…

In 1954 B. H. Neumann discovered that if G is a group in which all conjugacy classes are finite with bounded size, then the derived group G' is finite. Later (in 1957) Wiegold found an explicit bound for the order of G'. We study groups in…

Let $w$ be a word in $k$ variables. For a finite nilpotent group $G$, a conjecture of Amit states that $N_w(1) \ge |G|^{k-1}$, where $N_w(1)$ is the number of $k$-tuples $(g_1,...,g_k)\in G^{(k)}$ such that $w(g_1,...,g_k)=1$. Currently,…

For a finite group $G$, let $d(G)$ denote the probability that a randomly chosen pair of elements of $G$ commute. We prove that if $d(G)>1/s$ for some integer $s>1$ and $G$ splits over an abelian normal nontrivial subgroup $N$, then $G$ has…

We show that the set of all commuting probabilities in finite rings is a subset of the set of all commuting probabilities in finite nilpotent groups of class $\le2$. We believe that these two sets are equal; we prove they are equal, when…

The 'degree of k-step nilpotence' of a finite group G is the proportion of the tuples (x_1,...,x_{k+1}) in G^{k+1} for which the simple commutator [x_1,...,x_{k+1}] is equal to the identity. In this paper we study versions of this for an…

Let $G$ be a finite group, and let $\kappa(G)$ be the probability that elements $g$, $h\in G$ are conjugate, when $g$ and $h$ are chosen independently and uniformly at random. The paper classifies those groups $G$ such that $\kappa(G) \geq…

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.