Geodesic
A~new identity in the Lie ring of a~free group of prime exponent, and groups without the Hughes property
Izvestiya. Mathematics , Tome 29 (1987) no. 3, pp. 659-676.

Voir la notice de l'article provenant de la source Math-Net.Ru

A multilinear identity of degree $3p-2$ is given in explicit form, and it is shown that this identity holds in the associated Lie ring of a free group of prime exponent $p$. It is also shown that if this identity is not a consequence of the known identities of Wall of degree $2p-1$ and the $(p-1)$st Engel identity, there exists a finite $p$-group in which the index of the (nontrivial) Hughes subgroup is $p^3$. Bibliography: 13 titles. 
@article{IM2_1987_29_3_a7,
     author = {E. I. Khukhro},
     title = {A~new identity in the {Lie} ring of a~free group of prime exponent, and groups without the {Hughes} property},
     journal = {Izvestiya. Mathematics },
     pages = {659--676},
     publisher = {mathdoc},
     volume = {29},
     number = {3},
     year = {1987},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1987_29_3_a7/}
}
TY  - JOUR
AU  - E. I. Khukhro
TI  - A~new identity in the Lie ring of a~free group of prime exponent, and groups without the Hughes property
JO  - Izvestiya. Mathematics 
PY  - 1987
SP  - 659
EP  - 676
VL  - 29
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_1987_29_3_a7/
LA  - en
ID  - IM2_1987_29_3_a7
ER  - 
%0 Journal Article
%A E. I. Khukhro
%T A~new identity in the Lie ring of a~free group of prime exponent, and groups without the Hughes property
%J Izvestiya. Mathematics 
%D 1987
%P 659-676
%V 29
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_1987_29_3_a7/
%G en
%F IM2_1987_29_3_a7
E. I. Khukhro. A~new identity in the Lie ring of a~free group of prime exponent, and groups without the Hughes property. Izvestiya. Mathematics , Tome 29 (1987) no. 3, pp. 659-676. http://geodesic.mathdoc.fr/item/IM2_1987_29_3_a7/

[1] Magnus W., “A connection between the Baker–Hausdorff formula and a problem of Burnside”, Ann. of Math., 52:1 (1950), 111–126 | DOI | MR | Zbl

[2] Sanov I. Ya., “Ustanovlenie svyazi mezhdu periodicheskimi gruppami s periodom prostym chislom i koltsami Li”, Izv. AN SSSR. Ser. matem., 16:1 (1952), 23–58 | MR | Zbl

[3] Wall G. E., “On the Lie ring of a group of prime exponent”, Proc. Second Int. Conf. Theory of Groups (Canberra, 1973), Lecture Notes in Math., 372, Springer-Verlag, Berlin, London, 1974, 667–690 | MR

[4] Cannon J. J., “Some combinatorial and symbol manipulation programs in group theory”, Computational Problems in Abstract Algebra (Proc. Conf., Oxford, 1967), Pergamon, Oxford, 1970, 199–203 | MR

[5] Wall G. E., “On Hughes $H_p$-problem”, Proc. Int. Conf. Theory of Groups (Canberra, 1965), Gordon and Breach, New York, 1967, 357–362 | MR

[6] Xyxpo E. I., “O prisoedinennom koltse Li svobodnoi 2-porozhdennoi gruppy prostogo perioda i o gipoteze Khyuza dlya 2-porozhdennykh $p$-grupp”, Matem. sb., 118(160):4(8) (1982), 567–575 | MR

[7] Kostrikin A. I., “O svyazi mezhdu periodicheskimi gruppami i koltsami Li”, Izv. AN SSSR. Ser. matem., 21:3 (1957), 289–310 | MR

[8] Havas G., Wall G. E., Wamsley J. M., “The two generator restricted Burnside group of exponent five”, Bull. Austral. Math. Soc., 10 (1977), 459–470 | DOI | MR

[9] Wall G. E., “On the Lie ring of a group of prime exponent. II”, Bull. Austral. Math. Soc., 19:1 (1978), 11–29 | DOI | MR

[10] Xyxpo E. I., “O svyazi mezhdu gipotezoi Khyuza i sootnosheniyami v konechnykh gruppakh prostogo perioda”, Matem. sb., 116(158):2(10) (1981), 253–264 | MR

[11] Macdonald I. D., “Solution of the Hughes problem for finite $p$-groups of class $2p-2$”, Proc Amer. Math. Soc., 27:1 (1971), 39–42 | DOI | MR | Zbl

[12] Macdonald I. D., “The Hughes problem and others”, J. Austral. Math. Soc., 10 (1969), 475–479 | DOI | MR | Zbl

[13] Wall G. E., “Lie methods in group theory”, Topics in algebra (Proc. 18th Summer research institute of the Austral. Math. Soc., Canberra, 1978), Lecture Notes in Math., 697, Springer-Verlag, Berlin, New York, 1978, 137–173 | MR