On some problems of a statistical group-theory. IV (original) (raw)
Some Ascpects in probability group theory
2009
In this talk, we will give a brief survey of some aspects in the probability group theory. We will also focus on two old problems given by Erdos and Turan [1] on random generating set for a given group G, and Miller [2] on random commuting pair of elements in a finite group G. More precisely, for every finite group G, we would like to find the probability that the group generated by given two randomly chosen elements x and y to be a whole group. Moreover, we may state the similar probability when two elements of G are chosen randomly and independently commute. There are some known results given by some authors which will be given in the present talk and also state some recent researches in this area.
On the common transversal probability in finite groups
arXiv (Cornell University), 2022
. Let G be a finite group, and let H be a subgroup of G. We compute the probability, denoted by P G (H), that a left transversal of H in G is also a right transversal, thus a two-sided one. Moreover, we define, and denote by tp(G), the common transversal probability of G to be the minimum, taken over all subgroups H of G, of P G (H). We prove a number of results regarding the invariant tp(G), like lower and upper bounds, and possible values it can attain. We also show that tp(G) determines structural properties of G. Finally, several open problems are formulated and discussed.
Improved Upper Bounds on the Spreads of Some Large Sporadic Groups
2009
Let G be a group. We say that G has spread r if for any set of distinct elements {x1,..., xr}\subset G there exists an element y\in G with the property that <xi, y>=G for every 0<i<r+1. Few bounds on the spread of finite simple groups are known. In this paper we present improved upper bounds for the spread of many of the sporadic simple groups, in some cases improving on the best known upper bound by several orders of magnitude.
Annals of Mathematics, 2021
A group G is said to be 3 2-generated if every nontrivial element belongs to a generating pair. It is easy to see that if G has this property then every proper quotient of G is cyclic. In this paper we prove that the converse is true for finite groups, which settles a conjecture of Breuer, Guralnick and Kantor from 2008. In fact, we prove a much stronger result, which solves a problem posed by Brenner and Wiegold in 1975. Namely, if G is a finite group and every proper quotient of G is cyclic, then for any pair of nontrivial elements x1, x2 ∈ G, there exists y ∈ G such that G = x1, y = x2, y. In other words, s(G) 2, where s(G) is the spread of G. Moreover, if u(G) denotes the more restrictive uniform spread of G, then we can completely characterise the finite groups G with u(G) = 0 and u(G) = 1. To prove these results, we first establish a reduction to almost simple groups. For simple groups, the result was proved by Guralnick and Kantor in 2000 using probabilistic methods and since then the almost simple groups have been the subject of several papers. By combining our reduction theorem and this earlier work, it remains to handle the groups whose socles are exceptional groups of Lie type and this is the case we treat in this paper.
Random triangular groups at density
Compositio Mathematica, 2014
Let${\rm\Gamma}(n,p)$denote the binomial model of a random triangular group. We show that there exist constants$c,C>0$such that if$p\leqslant c/n^{2}$, then asymptotically almost surely (a.a.s.)${\rm\Gamma}(n,p)$is free, and if$p\geqslant C\log n/n^{2}$, then a.a.s.${\rm\Gamma}(n,p)$has Kazhdan’s property (T). Furthermore, we show that there exist constants$C^{\prime },c^{\prime }>0$such that if$C^{\prime }/n^{2}\leqslant p\leqslant c^{\prime }\log n/n^{2}$, then a.a.s.${\rm\Gamma}(n,p)$is neither free nor has Kazhdan’s property (T).
On a combinatorial problem in group theory
Israel journal of mathematics, 1993
Let n be a positive integer or infinity (denote ∞). We denote by W * (n) the class of groups G such that, for every subset X of G of cardinality n + 1, there exist a positive integer k, and a subset X 0 ⊆ X, with 2 ≤ |X 0 | ≤ n + 1 and a function f : {0, 1, 2,. .. , k} −→ X0, with f (0) = f (1) and non-zero integers t0, t1,. .. , t k such that [x t 0 0 , x t 1 1 ,. .. , x t k k ] = 1, where xi := f (i), i = 0,. .. , k, and xj ∈ H whenever x t j j ∈ H, for some subgroup H = D x t j j E of G. If the integer k is fixed for every subset X we obtain the class W * k (n). Here we prove that (1) Let G ∈ W * (n), n a positive integer, be a finite group, p > n a prime divisor of the order of G, P a Sylow p-subgroup of G. Then there exists a normal subgroup K of G such that G = P × K. (2) A finitely generated soluble group has the property W * (∞) if and only if it is finite-by-nilpotent. (3) Let G ∈ W * k (∞) be a finitely generated soluble group, then G is finite-by-(nilpotent of k-bounded class).