CHAPTER 2 Definition Throughout this article, we will use the following notation. If S is a group, then we define 5* := S\ {neutral element}. If S is a set which contains an element called "0", then we define S* := S \ {0}. It will always be clear from the context which definition we mean. Consider an abelian group (V,+) and a (possibly non-abelian) group (W, EB). The inverse of an element w G W will be denoted by 3w, and by WIBHJ 2 , we mean wi E B (E\w2). Suppose that there is a map ry from V x W to V and a map TW from W x V to W, both of which will be denoted by • or simply by juxtaposition, i.e. rv(v,w) — vw — v - w and rw(w,v) = wv = w • v for all v G V and all w G W. Consider a map F from V x V to W and a map i l from W x W to V, both of which are "bi-additive" in the sense that F(v! + v2lv) = F(Vl,v) E B F(v2,v) F(v, vi + v2) = F(u, vi) ffl F(v, v2) H(wi E B 1^2, w) = H(wi, w) + H(w2, w) if (io, t^i E B ^2) = #(w, wi) + if (w , ^2) for all v,vi,v2 G V and all w,wi,w2 G W. Suppose furthermore that there exists a fixed element e G V* and a fixed element ( G W*, and suppose that, for each v G V*, there exists an element v _ 1 G V*, and for each w G VK*, there exists an element K(W) G W*, such that, for all w,wi,w2 G VF and all v,vi,v2 G V, the following axioms are satisfied. We define Rad(F) Rad(#) Im(F) Im(ff) = eF(e,v) — v = {veV I F(v,y) = 0} = {w E ^ I #(w,W0 =0} = F(V,V) = H{W1W) (Qx) we = w. (Q2) ^ = ^. (Q3) (^1 ffl ^2)^ = ^ 1 ^ ffl U^- (Q4) (^1 +v2)w = viw + v2w. (Q5) M~ e ) • ^ = ^ ( - v ) . (Qe) v • w(—e) — vw. (Q7) Im(F) C Rad(if). (Q8) [WI,W2V]E = F(H(W2,WI),V). (Q9) ( G Rad(fi). (Q10) If Rad(F) / 0, then e G Rad(F). 3

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