A non-abelian 2-group whose endomorphisms generate a ring, and other examples of E-groups
Groups for which the distributively generated near-ring generated by the endomorphisms is in fact a ring are known as E-groups and are discussed in (3). R. Faudree in (1) has given the only published examples of non-abelian E-groups by presenting defining relations for a family of p–groups. However, as shown in (3), Faudree's group does not have the desired property when p = 2.
The Group of Automorphisms of a Distributively Generated Near Ring
S. D. Scott has shown that the group of automorphisms of the near ring generated by the automorphisms of a given group is isomorphic to the automorphism group of the given group if that group's automorphism is complete. Here that theorem is generalized by showing that the group of automorphisms of a near ring distributively generated by its units is a subgroup of the group of automorphisms of the group of units. The results obtained are used to find the automorphism groups of certain near rings. 1. Preliminaries. In this paper all near rings considered will be distributively generated (d.g.) and will contain an identity element. Also, they will be written as left-distributive near rings. If K represents an algebraic system, then Aut( K) (Inn(K)) will denote the group of automorphisms (inner automorphisms) of K. If V is a group, then A( V) will denote the d.g. near ring generated by the automorphisms of V. The following definitions are as given in [10]. For a near ring N, D(N) designates the (multiplicative) group of distributive units of N. If ti is in D(N), then an inner automorphism of N is defined by taking a to ,u-'a`a for all a in N. The center of N, denoted by Z(N), is the set of all ,s in D(N) such that y-ray = a for all a in N. N is said to be complete if Inn(N) = Aut( N) and Z( N) is trivial. Although it was not noted in [10], it is easy to show that if N is d.g. with S as the generating set and if js is in the center of S, then p is in Z(N) so that, in fact, Z(N) = Z(S). If H is a subset of an additive group, then -H ={-h I h in H}. For near ring terminology not defined here or in [10], see [8]. 2. Results. In Theorem 4 of [10] S. D. Scott gives an interesting statement about certain near ring automorphisms. However, behind Scott's theorem, there is a more general result. This generalization is incorporated in the next proposition. PROPOSITION 1. Let N be a near ring distributively generated by the set D of its distributive units. Then Aut(N) is embedded in Aut(D). In particular, if V is an arbitrary group such that Aut(V) = D(A(V)), then Aut(A(V)) < Aut(Aut(V)). If ,B is an automorphism of (N, +) and if /3, restricted to (D, *) is an automorphism of (D, -), then /8 is an automorphism of N. Received by the editors August 23, 1982. Presented at the 89th Annual Meeting of the American Mathematical Society in Denver, January 5-9, 1983. 1980 Mathematics Subject Classification. Primary 16A76; Secondary 20F28. ?1983 American Mathematical Society 0002-9939/82/0000-1042/$02.00
When to Cross the Spread? Trading in Two-Sided Limit Order Books
In this paper the problem of optimal trading in illiquid markets is addressed when the deviations from a given stochastic target function describing, for instance, external aggregate client flow are penalized. Using techniques of singular stochastic control, we extend the results of [F. Naujokat and N. Westray, Math. Financ. Econ., 4 (2011), pp. 299--335] to a two-sided limit order market with temporary market impact and resilience, where the bid ask spread is now also controlled. In addition to using market orders, the trader can also submit orders to a dark pool. We first show existence and uniqueness of an optimal control. In a second step, a suitable version of the stochastic maximum principle is derived which yields a characterization of the optimal trading strategy in terms of a nonstandard coupled forward-backward stochastic differential equation (FBSDE). We show that the optimal control can be characterized via buy, sell, and no-trade regions. The new feature is that we now get a nondegenerate no-...