Theory and Aplications of Finite Groups

Theory and Aplications of Finite Groups

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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1916 edition. Excerpt: ...Relating to the Groups of Isomorphisms. Every abelian group can be extended so that we obtain a group of twice the order of the original group, by means of operators of order 2 which transform each operator of this abelian group into its inverse. These groups may be regarded as a direct generalization of the dihedral groups, and may therefore be called generalized dihedral groups as regards Cf. Philosophical Magazine, vol. 231 (1908), p. 223. the given abelian groups. If the given abelian subgroup involves operators whose order exceeds 2, the corresponding general dihedral group is non-abelian and vice versa. Let G be any non-abelian generalized dihedral group of order g and let H be the abelian subgroup of order g/2 which was extended to obtain G. In any automorphism of G the g/2 non-invariant operators of order 2 must correspond to themselves, and hence the I oi G can be represented as a substitution group of degree h, h being the order of H. It is evident that the non-invariant operators of order 2 in G can be arranged in h different ways after the automorphism of H has been fixed. Hence the order of the / of G is the same as the order of the holomorph of H. We proceed to prove that the / of G is simply isomorphic with the holomorph of H. In fact, this / can be represented as a transitive substitution group of degree h which involves an invariant regular subgroup which is simply isomorphic with H, since G can be made simply isomorphic with itself in such a way that the operators of H correspond to themselves while the remaining operators of G correspond to themselves multiplied by an arbitrary operator of H. These isomorphisms therefore correspond to a regular subgroup of order h in I, I being represented on letters corresponding to more

Product details

  • Paperback | 116 pages
  • 189 x 246 x 6mm | 222g
  • Miami Fl, United States
  • English
  • black & white illustrations
  • 1236560450
  • 9781236560452