On counterexamples to the Hughes conjecture.
Let \(G\) be a finite group and \(p\) a prime. The \(p\)-Hughes subgroup \(H_p(G)\) is generated by all the elements of \(G\) which do not have order \(p\). The Hughes problem (1957) asks whether \(|G:H_p(G)|=p\) whenever \(1\neq H_p(G)\neq G\). It was solved in the affirmative for \(G\) being not a \(p\)-group [\textit{D. R. Hughes} and \textit{J. G. Thompson}, Pac. J. Math. 9, 1097-1101 (1959; Zbl 0098.25201)], as well as for some classes of \(p\)-groups. But \textit{G. E. Wall} [Proc. Int. Conf. Theory Groups, Canberra 1965, 357-362 (1965; Zbl 0189.31701)] constructed a 3-generator counterexample (``anti-Hughes \(p\)-group'') with \(|G:H_p(G)|=p^2\) for \(p=5\). Later the reviewer [Mat. Sb., N. Ser. 116(158), 253-264 (1981; Zbl 0473.20016)] showed that such an anti-Hughes \(p\)-group exists whenever a certain Wall relation \(W_{2p-1}\) in the associated Lie ring \(L(B(p))\) of the free countably generator group of exponent \(p\) is not a consequence of the \((p-1)\)-Engel relation mod \(p\). Furthermore, the reviewer [Mat. Sb., N. Ser. 118(160), 567-575 (1982; Zbl 0484.20017)] showed that even a 2-generator counterexample (giving answers to a series of other questions about finite \(p\)-groups) exists if another Wall relation \(W_{2p}\) is ``new'' in the same sense. The reviewer (ibid.) used computer-aided calculations to verify that this is the case for \(p=7\), thus constructing a 2-generator anti-Hughes group for \(p=7\). In this paper the authors use computer-aided calculations to produce explicit power-commutator presentations of some 3- and 2-generator anti-Hughes groups with \(|G:H_p(G)|=p^2\), including relatively small examples with orders \(5^{46}\) and \(7^{66}\). Other calculations in \(L(B(p))\) confirm that \(W_{2p-1}\) is ``new'' for \(p=11\), 13, 17, 19, thus giving the existence of 3-generator anti-Hughes groups for these primes. (It is of course expected that this is so for all primes larger than 3.) They also produce a 2-generator anti-Hughes group for \(p=11\). However, it is still an open question whether there exist anti-Hughes groups with \(|G:H_p(G)|=p^3\). In fact, \textit{G. E. Wall} [J. Algebra 104, 1-22 (1986; Zbl 0603.20017)] showed that an anti-Hughes group with \(|G:H_p(G)|=p^k\) exists if \textit{M. R. Vaughan-Lee}'s [Bull. Lond. Math. Soc. 17, 113-133 (1985; Zbl 0544.20032)] multilinear relations in \(L(B(p))\) of weights \(1+i(p-1)\) are ``new'' for each \(i=1,2,\dots,k\). The authors verified that this is not the case for \(k=3\) and \(p=5\). But the next interesting case of \(k=3\) and \(p=7\) still remains out of reach of modern computers -- which is rather amazing given that the previous important steps of showing that \(W_{2p-1}\) for \(p=5\) and \(W_{2p}\) for \(p=7\) are ``new'' were made in 1965 and 1982, respectively! Finally, it is worth mentioning that, in spite of counterexamples, the reviewer showed [Algebra Logic 26, No. 5, 398-401 (1987); translation from Algebra Logika 26, No. 5, 642-646 (1987; Zbl 0658.20015)] that the Hughes problem has affirmative solution for almost all finite \(p\)-groups, in a certain precise sense.
scientific article; zbMATH DE number 5610505
scientific article