Multihomomorphisms from Groups into Groups of Real Numbers

P. Youngkhong, K. Savettaraseranee

Abstract

By a multihomomorphism from a group G into a group G' we mean a multifunction f from G into G' such that

f(xy) = f(x)f(y) $({st : s \in f(x) and t \in f(y)})$

for all $x, y \in G$. We denote by MHom(G,G') the set of all multihomomorphisms from G into G'. It is shown that if $f \in$ MHom(G,G') where G' is a subgroup of (R, +), then either f is a homomorphism or there is an infinite cardinal number $\eta$ such that $ |f(x)| = \eta $ for all $x \in G$. If $f \in $ MHom(G,G') where G' is a subgroup of $(R*,\cdot )$, then (i) f is a homomorphism, (ii) |f(x)|= 2 for all $ xor (iii) there is an infinite cardinal number $\eta$ such that $ |f(x)| = \eta$ .