Center (group Theory) - Examples

Examples

  • The center of an abelian group G is all of G.
  • The center of the Heisenberg group G are all matrices of the form :\begin{pmatrix} 1 & 0 & z\\ 0 & 1 & 0\\ 0 & 0 & 1\\ \end{pmatrix}
  • The center of a nonabelian simple group is trivial.
  • The center of the dihedral group Dn is trivial when n is odd. When n is even, the center consists of the identity element together with the 180° rotation of the polygon.
  • The center of the quaternion group is .
  • The center of the symmetric group Sn is trivial for n ≥ 3.
  • The center of the alternating group An is trivial for n ≥ 4.
  • The center of the general linear group is the collection of scalar matrices .
  • The center of the orthogonal group is .
  • The center of the multiplicative group of non-zero quaternions is the multiplicative group of non-zero real numbers.
  • Using the class equation one can prove that the center of any non-trivial finite p-group is non-trivial.
  • If the quotient group is cyclic, G is abelian (and so G = Z(G), and is trivial).
  • The quotient group is not isomorphic to the quaternion group .

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