Factorizations of finite groups (original) (raw)

On factorizations of finite groups

2021

Let G be a finite group and let {A1, . . . , Ak} be a collection of subsets of G such that G = A1 . . . Ak is the product of all the Ai and card(G) = card(A1) . . . card(Ak). We shall write G = A1 · . . . · Ak, and call this a kfold factorization of the form (card(A1), . . . , card(Ak)). We prove that for any integer k ≥ 3 there exist a finite group G of order n and a factorization of n = a1 . . . ak into k factors other than one such that G has no k-fold factorization of the form (a1, . . . , ak).

Classification of Fundamental Groups of Galois Covers of Surfaces of Small Degree Degenerating to Nice Plane Arrangements

Arxiv preprint arXiv: …, 2010

Let X be a surface of degree n, projected onto CP 2. The surface has a natural Galois cover with Galois group S n. It is possible to determine the fundamental group of a Galois cover from that of the complement of the branch curve of X. In this paper we survey the fundamental groups of Galois covers of all surfaces of small degree n ≤ 4, that degenerate to a nice plane arrangement, namely a union of n planes such that no three planes meet in a line. We include the already classical examples of the quadric, the Hirzebruch and the Veronese surfaces and the degree 4 embedding of CP 1 ×CP 1 , and also add new computations for the remaining cases: the cubic embedding of the Hirzebruch surface F 1 , the Cayley cubic (or a smooth surface in the same family), for a quartic surface that degenerates to the union of a triple point and a plane not through the triple point, and for a quartic 4-point. In an appendix, we also include the degree 8 surface CP 1 ×CP 1 embedded by the (2, 2) embedding, and the degree 2n surface embedded by the (1, n) embedding, in order to complete the classification of all embeddings of CP 1 × CP 1 , which was begun in [23]. Partially supported by the Emmy Noether Research Institute for Mathematics (center of the Minerva Foundation of Germany), the Excellency Center "Group Theoretic Methods in the Study of Algebraic Varieties" of the Israel Science Foundation, and EAGER (EU network, HPRN-CT-2009-00099).

A Remark on Generalized Covering Groups

2011

Let calNc{\cal N}_ccalNc be the variety of nilpotent groups of class at most c(cgeq2)c\ \ (c\geq 2)c(cgeq2) and G=ZroplusZsG=Z_r\oplus Z_s G=ZroplusZs be the direct sum of two finite cyclic groups. It is shown that if the greatest common divisor of rrr and sss is not one, then GGG does not have any calNc{\cal N}_ccalNc-covering group for every cgeq2c\geq 2cgeq2. This result gives an idea that Lemma 2 of J.Wiegold [6] and Haebich's Theorem [1], a vast generalization of the Wiegold's Theorem, can {\it not} be generalized to the variety of nilpotent groups of class at most cgeq2c\geq 2cgeq2.

Finite groups of small genus

2015

For a finite group G, the Hurwitz space H in r,g (G) is the space of genus g covers of the Riemann sphere with r branch points and the monodromy group G.

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,, .

On the factorization numbers of some finite $ p $-groups

This note deals with the computation of the factorization number F 2 (G) of a finite group G. By using the Möbius inversion formula, explicit expressions of F 2 (G) are obtained for two classes of finite abelian groups, improving the results of Factorization numbers of some finite groups, Glasgow Math. J. (2012).

Doubly Hurwitz Beauville groups

Cornell University - arXiv, 2017

If S is a Beauville surface (C 1 × C 2)/G, then the Hurwitz bound implies that |G| ≤ 1764 χ(S), with equality if and only if the Beauville group G acts as a Hurwitz group on both curves C i. Equivalently, G has two generating triples of type (2, 3, 7), such that no generator in one triple is conjugate to a power of a generator in the other. We show that this property is satisfied by alternating groups A n , their double covers 2.A n , and special linear groups SL n (q) if n is sufficiently large, but by no sporadic simple groups or simple groups L n (q) (n ≤ 7), 2 G 2 (3 e), 2 F 4 (2 e), 2 F 4 (2) ′ , G 2 (q) or 3 D 4 (q) of small Lie rank.