The Waring Problem for Finite Quasisimple Groups. II (original) (raw)

Effective results on the Waring problem for finite simple groups

American Journal of Mathematics, 2015

Let G be a finite quasisimple group of Lie type. We show that there are regular semisimple elements x, y ∈ G, x of prime order, and |y| is divisible by at most two primes, such that x G • y G ⊇ G \ Z(G). In fact in all but four cases, y can be chosen to be of square-free order. Using this result, we prove an effective version of one of the main results of [LST1] by showing that, given any integer m ≥ 1, if the order of a finite simple group S is at least m 8m 2 , then every element in S is a product of two m th powers. Furthermore, the verbal width of x m on any finite simple group S is at most 80m 2 log 2 m + 56. We also show that, given any two non-trivial words w 1 , w 2 , if G is a finite quasisimple group of large enough order, then w 1 (G)w 2 (G) ⊇ G \ Z(G).

A product decomposition for the classical quasisimple groups

Journal of Group Theory, 2007

We prove that every quasisimple group of classical type is a product of boundedly many conjugates of a quasisimple subgroup of type An. 1 Main Result and Notation Let S be a quasisimple group of classical Lie type X , that is one from {A n , B n , C n , D n , 2 A n , 2 D n }. Its classical definition is as a quotient of some group of linear transformations of a vector space over a finite field F preserving a nondegenerate form. We shall not make use of this geometry but instead rely on the Lie theoretic approach. In other words we view S as the group of fixed points of certain automorphism of an algebraic group defined as follows:

Words with Few Values in Finite Simple Groups

The Quarterly Journal of Mathematics, 2013

We construct words with small image in a given finite alternating or unimodular group. This shows that word width in these groups is unbounded in general. Let w be a group word, i.e., an element of the free group on x 1 ,. .. , x d. For a group G we denote the set of values of w by G w := {w(g 1 ,. .. , g d) ±1 | g i ∈ G} and the verbal subgroup G w is w(G). The study of the images of the word maps dates back to the theory of varieties of groups (see [1]) and the work of P. Hall and his students, for a modern exposition also see [7]. Recently, there has been a lot of progress in understanding G w when G is a finite group, in particularly a finite simple group. In [3] it was proved that when G is a finite simple group then every element of G is a commutator and in addition every element of G is a product of two squares [4]. More generally [2] shows that for any w we have G = G w G w when G is a sufficiently large finite simple group. In this note, we show that the requirement of the size of G can not be removed, even if we only require that G = (G w) k for a fixed k. Theorem 1. For any k there exist a word w and a finite simple group G, such that w is not an identity in G, but G = (G w) k. We obtain this as an immediate corollary of the following results about alternating groups and special linear groups.

The Word Problem for Finitely Generated Soluble Groups of Finite Rank

Bulletin of the London Mathematical Society, 1984

THEOREM 2. For each prime p and each p-adic integer a there is a 4-generator centre by metabelian group G(a) with finite rank whose centre is of type p°°. The group G(a) has soluble word problem if and only if a. is computable.

On some conjectures related to finite nonabelian simple groups

arXiv: Group Theory, 2020

In this note we provide some counterexamples for the conjecture of Moret\'{o} on finite simple groups, which says that any finite simple group GGG can determined in terms of its order ∣G∣|G|∣G∣ and the number of elements of order ppp, where ppp the largest prime divisor of ∣G∣|G|∣G∣. Moreover, we show that this conjecture holds for all sporadic simple groups and alternating groups AnA_nAn, where nneq8,10n\neq 8, 10nneq8,10. Some related conjectures are also discussed.

On finite factorizable groups*1

Journal of Algebra, 1984

ON FINITE FACTORIZABLE GROUPS 523 (I) A, with r > 5 a prime and A N A,-, . (II) M,, and either A is solvable or A N M,,. (III) M,, and either B is Frobenius of order 11 . 23 or B is cyclic of order 23 and A N M,, .

A computer study of the orders of finite simple groups

Mathematics of Computation, 1968

In this paper we describe how several results of finite group theory were applied in a computer study of the distribution of the orders of finite simple groups. Besides being of some small interest in themselves, the results obtained here may hopefully give some hint as to what might be proved in general about the distribution of finite simple group orders.