Hostname: page-component-848d4c4894-pjpqr Total loading time: 0 Render date: 2024-07-05T20:30:17.104Z Has data issue: false hasContentIssue false
  • English
  • Français

The derived group of a 2-group

Published online by Cambridge University Press:  24 October 2008

Norman Blackburn
Norman Blackburn
Affiliation:
Department of Mathematics, University of Manchester For Bertram Huppert, on his sixtieth birthday
Get access Rights & Permissions [Opens in a new window]

Extract

Burnside[1] considered possible restrictions on the derived group G′ of p-group G and showed that if G′ is non-Abelian, the centre Z(G′) of G′ is not cyclic. This implies that |G′: G″| ≥ p3. Many other restrictions on G′ are to be found in Hall's famous paper [2], but in 1954 Hall proved that if p is odd and |G′: G″| = p3, then |G′| ≤ p. So far as I know, no proof of this is to be found in the literature, but it follows from the lemma below. Our concern here is with the case p = 2, and we shall prove the following.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 101 , Issue 2 , March 1987 , pp. 193 - 196
Copyright
Copyright © Cambridge Philosophical Society 1987

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Burnside, W.. On some properties of groups whose orders are powers of primes. Proc. London Math. Soc. (2) 11 (1912), 225–245.Google Scholar
[2]Hall, P.. A contribution to the theory of groups of prime-power order. Proc. London Math. Soc. (2) 36 (1933), 29–95.Google Scholar
[3]Blackburn, N.. On prime-power groups in which the derived group has two generators. Proc. Cambridge Philos. Soc. 53 (1957), 19–27.CrossRefGoogle Scholar