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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
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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

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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