Galois-theoretical groups

A group G is called Galois-theoretical if CGCA(H)=H for any subgroup H of G and CACG(B)=B for any subgroup B of A=Aut(G). This paper shows that a group G is Galois-theoretical if and only if G is isomorphic to the trivial group, to the cyclic group of order 3, or to the symmetric group of degree 3

Partition numbers of finite solvable groups

A group partition is a group cover in which the elements have trivial pairwise intersection. Here we define the partition number of a group - the minimal number of subgroups necessary to form a partition - and examine some of its properties, including its relation to the covering number for solvable groups

The Proportion of Fixed-Point-Free Elements of a Transitive Permutation Group

In 1990 Hendrik W. Lenstra, Jr. asked the following question: if G is a transitive permutation group of degree n and A is the set of elements of G that move every letter, then can one find a lower bound (in terms of n) for f(G) = |A|/|G|? Shortly thereafter, Arjeh Cohen showed that 1/n is such a bound. Lenstra’s problem arose from his work on the number field sieve. A simple example of how f(G) arises in number theory is the following: if h is an irreducible polynomial over the integers, consider the proportion: |{primes ≤ x | h has no zeroes mod p}| / |{primes ≤ x}| As x → ∞, this ratio approaches f(G), where G is the Galois group of h considered as a permutation group on its roots. Our results in this paper include explicit calculations of f(G) for groups G in several families. We also obtain results useful for computing f(G) when G is a wreath product or a direct product of permutation groups. Using this we show that {f(G) | G is transitive} is dense in [0, 1]. The corresponding conclusion is true if we restrict G to primitive groups

Groups, transversals, and loops

summary:A family of loops is studied, which arises with its binary operation in a natural way from some transversals possessing a ``normality condition''

Finite groups with a special 2-generator property, and order of centralizers in finite groups

This paper deals with finite groups, and has two parts. In part I J. L. Brenner and James Wielgold (I,3) defined a finite nonabelian group G as lying in \Gamma\sb1\sp{(2)} (spread one-two) if for every 1 $\not=$ x $\in$ G, either x is an involution and G = $\langle$x,y$\rangle$ for some y $\in$ G or x is not an involution and there is an involution z $\in$ G with G = $\langle$x,z$\rangle$. We show that "most" of the simple groups of Lie type do not lie in \Gamma\sb1\sp{(2)}, we classify all those solvable groups which lie in \Gamma\sb1\sp{(2)}, and we show that a finite non-simple non-solvable group lies in \Gamma\sb1\sp{(2)} if it is isomorphic to the semi-direct product of N and $\langle$x$\rangle$ where x is an involution and N is a simple nonabelian group. Many simple groups are excluded from being candidates for the N above.Part II includes a characterization of all groups G having a subgroup A with $\vert$A$\Vert$C\sb{\rm G}(A)$\vert$ $>$ $\vert$G$\vert$, and those for which m\sb1 = sup $\{\vert$B$\Vert$C\sb{\rm G}(B)$\vert$: B $\le$ G$\}$ = $\vert$G$\vert$. It is shown also that if G is not a direct product, then either there exists a nontrivial characteristic abelian subgroup A of G with $\vert$A$\Vert$C\sb{\rm G}(A)$\vert$ $\ge$ $\vert$G$\vert$, or $\vert$B$\Vert$C\sb{\rm G}(B)$\vert$ $<$ $\vert$G$\vert$ for any proper nontrivial subgroup B of G.U of I OnlyETDs are only available to UIUC Users without author permissio

Decomposition of Groups into Twisted Subgroups and Subgroups

.This article is subsequent to our previous one, entitled Involutory decomposition of groups into twisted subgroups and subgroups [7]. The twisted subgroups resulting from the involutory decomposition of groups into twisted subgroups and subgroups in [7] turn out to be gyrocommutative gyrogroups. In contrast, the twisted subgroups resulting from the (non-involutory) decomposition of groups into twisted subgroups and subgroups that we present in this article need not be gyrocommutative. Twisted subgroups arise in the study of problems in computational complexity [1] and in the study of gyrogroups [7]. Gyrogroups are grouplike structures that first arose in the study of Einstein&apos;s velocity addition in the special theory of relativity [23, 24]. We showed in [7] that any gyrogroup is an extension of a group by a gyrocommutative gyrogroup. The gyrogroups that we construct in this article demonstrate that this extension is not trivial. x1. Introduction Gyrogroup theory is an algebraic theo..