Abstract
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.
Publisher
Cambridge University Press (CUP)
Reference3 articles.
1. A contribution to the theory of groups of prime-power order;Hall;Proc. London Math. Soc.,1933
2. On some properties of groups whose orders are powers of primes;Burnside;Proc. London Math. Soc.,1912
3. On prime-power groups in which the derived group has two generators
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