(a) Write the set of lines of PG(2,3) in set builder (not set roster) form and show that all these lines are distinct i.e. there are precisely 13 lines. Hint: solve some equation in Z 13- (b) Show that the cardinality of each line is 18;= 4. Hint: solve some equation in Z13. (C) For each point j, what are the lines g; on it? Hence show that the number of lines on a point of PG(2,3) is 4. (d) Show that Vi eP*:=P\{0}, 3!(d,e) E D X D s.t. d Ee and i = d e. (Do this by a suitable 4 x 4 Cayley table with the minus operation on D. (The "exists with exclamation mark means "exists a unique".) A finite projective plane PG(2,3) of order 3, can be defined as follows: The points are elements of the ring Z13:= {0,1,...,12}where these are taken modulo 13 (the remainder when an integer is divided by 13). Let D be {0,1,3,9} CP:=Z13 and let the lines be subsets gi=i+D (mod 13). Don't draw the whole structure PG(2,3)! Writing down all the lines as sets is possible. (C) For each point j, what are the lines g; on it? Hence show that the number of lines on a point of PG(2,3) is 4. (d) Show that Vi eP*:= P\{0}, 3! (d,e) E D D s.t. d #e and i = d - e. (Do this by a suitable 4 x 4 Cayley table with the minus operation on D. (The exists with exclamation mark" means "exists a unique".) (a) Write the set of lines of PG(2,3) in set builder (not set roster) form and show that all these lines are distinct i.e. there are precisely 13 lines. Hint: solve some equation in Z 13- (b) Show that the cardinality of each line is 18;= 4. Hint: solve some equation in Z13. (C) For each point j, what are the lines g; on it? Hence show that the number of lines on a point of PG(2,3) is 4. (d) Show that Vi eP*:=P\{0}, 3!(d,e) E D X D s.t. d Ee and i = d e. (Do this by a suitable 4 x 4 Cayley table with the minus operation on D. (The "exists with exclamation mark means "exists a unique".) A finite projective plane PG(2,3) of order 3, can be defined as follows: The points are elements of the ring Z13:= {0,1,...,12}where these are taken modulo 13 (the remainder when an integer is divided by 13). Let D be {0,1,3,9} CP:=Z13 and let the lines be subsets gi=i+D (mod 13). Don't draw the whole structure PG(2,3)! Writing down all the lines as sets is possible. (C) For each point j, what are the lines g; on it? Hence show that the number of lines on a point of PG(2,3) is 4. (d) Show that Vi eP*:= P\{0}, 3! (d,e) E D D s.t. d #e and i = d - e. (Do this by a suitable 4 x 4 Cayley table with the minus operation on D. (The exists with exclamation mark" means "exists a unique".)
