On subgroups of finite \(p\)-groups
The paper ``Generalizations of certain elementary theorems on \(p\)-groups'' by \textit{N.~Blackburn} [Proc. Lond. Math. Soc., III. Ser. 11, 1-22 (1961; Zbl 0102.01903)] has been very influential in the theory of (finite) \(p\)-groups and their applications. For instance, Blackburn's classification of \(p\)-groups that do not contain an elementary Abelian normal subgroup of order \(p^3\) has been instrumental in the Odd Order paper of \textit{W.~Feit} and \textit{J.~G.~Thompson} [Pac. J. Math. 13, 775-1029 (1963; Zbl 0124.26402)]. The paper under review may be considered to some extent a present day version of the paper of Blackburn, and is concerned with various results in which the structure of ``small'' subgroups of a \(p\)-group \(G\) influences the structure of \(G\). For instance, in Section~6 a new, short proof of the results of Blackburn mentioned above is given. In No.~2, Part I, Chapter G of their ``The classification of the finite simple groups'' series [Mathematical Surveys and Monographs. Vol. 40, No.~2. Providence, RI: Am. Math. Soc. (1996; Zbl 0836.20011)], \textit{D.~Gorenstein, R.~Lyons} and \textit{R.~Solomon} show (Lemma~11.18) that if an involution \(t\) acts on a \(p\)-group \(G\) of odd order, and if \(G\) has an elementary Abelian subgroup of order \(p^3\), then \(t\) normalizes one such subgroup. This is extended here (Theorem~1) to cover elementary Abelian subgroups of order \(p^4\). Another result from the book by Gorenstein et al.\ is given here a cohomology-free proof, and this is used to count elements of order \(p\) in a Sylow \(p\)-subgroup of a symmetric group of degree \(p^n\). Among the many other results of this paper, we may mention Theorem~4, which improves upon a result of \textit{P.~Hall} (Theorem~2.64 of [Proc. Lond. Math. Soc., II. Ser. 40, 468-480 (1935), 481-501 (1936; Zbl 0015.20201)]), and states that if \(G\) is a \(p\)-group, and \(k>1\) is an integer, then one of the following holds: (a) the subgroup generated by the elements of order at most \(p^k\) has exponent at most \(p^k\); (b) \(G\) is of maximal class and order \(p^{2+(p-1)k}\); (c) \(G\) has a subgroup of order \(p^{2+(p-1)k}\) and exponent at most \(p^k\). Spoiler: Blackburn's paper referred to above is quoted in the bibliography as ``Generalizations of certain \textit{elementary Abelian} theorems on \(p\)-groups''.
scientific article; zbMATH DE number 1428239
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