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 the 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 , then 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 d 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 y 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~ l q~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 t Y i s a normal subgrou p o f X. B y a line, plane, hyperplane i n a finite ^-dimensiona l vecto r 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 http://dx.doi.org/10.1090/cbms/022/01
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