Prerequisites and Notation
We assume a knowledge o f th e basi c facts abou t sets , groups, fields an d vector spaces .
If X an d Y ar e sets, then po w X wil l denote th e se t o f al l subsets of X; X C Y
will denote stric t inclusion ; X - Y wil l denote th e differenc e set ; X -* Y wil l denot e a
surjection, I ~ y a n injection, X - * Y a bijection, an d I ^ - ^ a n arbitrary mapping.
If f:X--Y i s a mapping an d Z i s a subset o f X, i.e. , Z i s an element o r point i n
pow X, the n fZ i s the subse t {fz\z G Z} o f Y\ thi s provides a natural extensio n o f / :
X -- Y t o / : po w X •- pow Y
y
namel y th e on e obtaine d b y sendin g Z t o fZ fo r al l
Z i n po w X; i f / i s respectively injective , surjective, bijective , the n so is its extension to th
power sets.
By the natural number s we mean th e number s 1, 2, 3, •• .
If X i s any additiv e group , in particula r i f X i s either a field or a vector space , the
X wil l denote th e se t o f nonzer o element s o f X\ i f X i s a field, the n X i s to b e regarde
as a multiplicative group . Us e F fo r th e finit e fiel d o f q elements . I f X i s an arbitrar
multiplicative grou p an d Y i s a nonempty subse t o f X, w e use Y) fo r th e grou p gener
ated by Y; ce n X fo r th e cente r o f X\ \p, q] fo r th e commutato r pqp~ lq~x o f ele-
ments p, q o f X; DX fo r th e commutato r subgrou p o f X ; an d Y X t o indicat e tha
Y i s a normal subgrou p o f X. B y a line, plane, hyperplane i n a finite ^-dimensiona l vecto
space V w e mean a subspace o f dimensio n 1, 2, n - 1, respectively . W e use V fo r th e
dual space of V; fo r th e annihilato r i n V o f a subset S o f V\ an d fo r th e
annihilator i n V o f a subset T of V. Fo r an y A E F let (a) denot e th e line Fa.
V wil l always denote a n ^-dimensiona l vecto r spac e ove r a field F wit h 1 n °
and V
x
wil l always denote a n n
x
-dimensional vecto r spac e ove r a field F
x
wit h 1 n
x
1
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