Wreath Product - Properties

Properties

  • Since the finite direct product is the same as the finite direct sum of groups it follows that the unrestricted A WrΩ H and the restricted wreath product A wrΩ H agree if the H-set Ω is finite. In particular this is true when Ω = H is finite.
  • A wrΩ H is always a subgroup of A WrΩ H.
  • Universal Embedding Theorem: If G is an extension of A by H, then there exists a subgroup of the unrestricted wreath product A≀H which is isomorphic to G.
  • If A, H and Ω are finite, then
|A≀ΩH| = |A||Ω||H|.

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