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
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 ( 5 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
= 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~
) ^ = ^ ( - v ) .
(Qe) v w(—e) vw.
(Q7) Im(F) C Rad(if).
(Q8) [WI,W2V]E = F(H(W2,WI),V).
(Q9) ( 5 G Rad(fi).
(Q10) If Rad(F) / 0, then e G Rad(F).
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