“Woah! It's like Spotify but for academic articles.”

Instant Access to Thousands of Journals for just $40/month

Get 2 Weeks Free

The direct extension theorem

The direct extension theorem J. Group Theory 9 (2006), 307­316 DOI 10.1515/JGT.2006.020 ( de Gruyter 2006 Joseph Ayoub (Communicated by R. M. Guralnick) 1 Introduction Let H and K be finite groups and consider an extension 1 ! H ! G ! K ! 1. In general the isomorphism class of G does not determine the extension. The main result of this paper is it does in the special case that G G H Â K. Our theorem can be restated as follows: Theorem 1.1. Let G ¼ H Â K be a finite group. Suppose that H0 is a normal subgroup of G and assume that H0 G H and G=H0 G G=H. Then H0 is a direct factor of G (i.e. G ¼ H0 Â K0 for some complement K0 ). The corresponding result for finitely generated modules over a noetherian commutative ring is well known (see [2] or [3]). In particular, the result holds for finitely generated abelian groups. 2 Preliminary results In this section, G is any finite group, not necessarily the group that appears in the theorem. 2.1 Subgroups of a direct product G F H D K. We shall often use the following elementary results without http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Group Theory de Gruyter
Loading next page...
 
/lp/de-gruyter/the-direct-extension-theorem-Rl1zroqZeL

You're reading a free preview. Subscribe to read the entire article.

And millions more from thousands of peer-reviewed journals, for just $40/month

Get 2 Weeks Free

To be the best researcher, you need access to the best research

  • With DeepDyve, you can stop worrying about how much articles cost, or if it's too much hassle to order — it's all at your fingertips. Your research is important and deserves the top content.
  • Read from thousands of the leading scholarly journals from Springer, Elsevier, Nature, IEEE, Wiley-Blackwell and more.
  • All the latest content is available, no embargo periods.

Stop missing out on the latest updates in your field

  • We’ll send you automatic email updates on the keywords and journals you tell us are most important to you.
  • There is a lot of content out there, so we help you sift through it and stay organized.