1 The Heisenberg group Let p be an odd prime The Heisenberg group over Z pZ is defined as 1 a c Hp 0 1 b a , b , c in Z pZ 0 0 1 The operation is matrix multiplication mod p We name its generators 1 0 1 1 0 0 1 1 0 x 0 1 0 , y 0 1 1 , z 0 1 0 , 0 0 1 0 0 1 0 0 1 so that 1 0 0 a 1 0 c b y b z c xa 1 From this relation it is clear that Hp p 3 ( a ) Write Hp in terms of generators and relations ( b ) If G is any nonabelian group of order p 3 , can you prove that G Hp ( c ) Prove that Hp ( Z pZ Z pZ ) Z pZ

Question

1   The Heisenberg group  Let p be an odd prime  The Heisenberg group over Z   pZ is defined as 1 a c Hp     0 1 b   a , b , c in Z   pZ   0 0 1 The operation is matrix multiplication mod p   We name its generators 1 0 1 1 0 0 1 1 0 x   0 1 0 , y   0 1 1 , z   0 1 0 , 0 0 1 0 0 1 0 0 1 so that 1 0 0 a 1 0 c b   y b z c xa   1 From this relation it is clear that   Hp     p 3   ( a ) Write Hp in terms of generators and relations  ( b ) If G is any nonabelian group of order p 3 , can you prove that G   Hp   ( c ) Prove that Hp   ( Z   pZ Z   pZ ) Z   pZ

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1 . The Heisenberg group. Let p be an odd prime. The Heisenberg group over Z / pZ is defined as 1 a c Hp

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