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.
Groups in which each element commutes with its endomorphic images
Let G be a finite group in which each element commutes with its endomorphic images. We will give a counterexample to the conjecture that G is abelian. Let G be a finite group in which each element commutes with its endomorphic images. We will give a counterexample to the conjecture that G is abelian. For g and h in G, g and g-1(h-1lgh) commute. Thus G satisfies the identical relation (x, y, x) = 1. Therefore G satisfies the identical relations (x, y, z, w)=1 and (x, y, z)3=I (see [1, p. 322]). Hence G is nilpotent of class <3 and if G is a p-group for a prime ps3, G is nilpotent of class <2. We will exhibit for any prime p a nilpotent class 2 p-group in which each element commutes with all of its endomorphic images. Let p2 G = (a,, a2, a3, a4:a = 1, (ai, aj, ak) = 1, (1 < i,j, k < 4) p p ~~~~~~~~p and (1) (a,, a2) = al, (2) (a,, a3) = a3, (3) (a1, a4) = a4, (4) (a2, a3) = a2, (5) (a2, a4) = 1, (6) (a3, a4) = a3). Let E(G) and A (G) denote the endomorphisms and automorphisms of G respectively. Let EZ(G) = {GCE(G):0(G) <Z(G) } and AZ(G) {OCA(G):O(g)g-1E'Z(G) for all gEEG } =central automorphisms of G. We will prove THEOREM. E(G) =EZ(G)kUAZ(G). COROLLARY. Each element in G commutes with its endomorphic images. LEMMA 1. G| =p PROOF. Clearly from the defining relations I GI <p8. We will construct a group of order p8 which satisfies the defining relations of G. Let Received by the editors February 11, 1970 and, in revised form, April 24, 1970. AMS 1969 subject classifications. Primary 2025, 2040.
Synthesis of quasi-stationary inhomogeneous impedance body of arbitrary shape
Formulas were obtained for the reflection factors of a plane wave from the inhomogeneous impedance plane, which reflects the incident wave in the given direction. The problem of synthesis of impedance inhomogeneous rotation body was solved for an arbitrary shaped generatrix for the given scattering diagram for co- and cross polarizations. Impedance distributions were obtained in explicit form. Restrictions on the class of implementable scat- terplots were formulated.