2 IÖUI S AUSLANDE R
and i f it is not abelian , it s commutator subgrou p ha s dimensio n one .
Let B = A+S, wher e A i s the alternating bilinea r for m A = j ( B -B ) ,
where th e super Script t denote s th e transpose o f th e for m B an d S is
the Symmetri e bilinea r for m S = ^(B+B ) . W e wil l sometime s als o us e th e
notation B = A(B)+S(B).
Now le t A b e an alternating bilinea r form , an d le t S b e a Symmetri e
bilinear form . W e ma y conside r N(A ) and N(A+S) . W e wil l no w provid e an
isomorphism betwee n N(A ) and N(A+S) . Le t a subscript 0 denot e tha t the
element i s in N(A) and a subscript 1 denot e tha t th e element i s in N(A+S) .
Thus
^l-WvVo= (W
W ^ i - V ^
and
(vl'Vl(v2'Vi =
VVVV^l'V^VV*!
Let 7r s{iv't)o) = (v t+^Slv v)\
Then it is a straightforward exercis e t o verify tha t
Tg : N(A)-N(A+S )
is a n isomorphism. A n elementary computatio n give s tha t
*"* : N(A+S) - N(A )
is give n b y n~ (v, t) = (v, - jS(v, v)+t ) . Henc e N(A+ S ) an d N(A+S
2
are
isomorphic unde r th e isomorphism j r o » ((v,t ) ) = (v, jS9 (v , v)-jS . (v, v)+t)„
S 2 S l l Z L
where th e subscript 2 denote s th e group N(A+S^ ) an d th e subscrip t 1 th e
group N(A+ S ).
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