n-capability of A-groups

Document Type : Original Scientific Paper

Authors

1 Department of Pure Mathematics, Ferdowsi University of Mashhad, P. O. Box 1159, Mashhad 91775, I. R. Iran

2 Department of Pure Mathematics, Payame Noor University, Tehran, I. R. Iran

Abstract

Following P. Hall a soluble group whose Sylow subgroups are all abelian is called A-group. The purpose of this article is to give a new and shorter proof for a criterion on the capability of A-groups of order p^2q, where p and q are distinct primes. Subsequently we give a sufficient condition for n-capability of groups having the property that their center and derived subgroups have trivial intersection, like the groups with trivial Frattini subgroup and A-groups. An interesting necessary and sufficient condition for capability of the A-groups of square free order will be also given.

Keywords


[1] R. Baer, Groups with preassigned central and central quotient groups, Trans. Amer. Math. Soc. 44 (1938) 387 - 412.
[2] F. R. Beyl, U. Felgner and P. Schmid, On groups occurring as center factor groups,
J. Algebra 61 (1979) 161 - 177.
[3] J. Burns and G. Ellis, On the nilpotent multipliers of a group, Math. Z. 226 (1997) 405 - 428.
[4] W. Burnside, Theory of Groups of Finite Order, 2nd, ed., Cambridge University Press, Cambridge, 1911.
[5] G. Ellis, On the capability of groups,
Proc. Edinb. Math. Soc. 41 (1998) 487 - 495.
[6] P. Hall, The classification of prime-power groups,
J. Reine Angew. Math. 182 (1940) 130 - 141.
[7] P. Hall, The construction of soluble groups,
J. Reine Angew. Math. 182 (1940) 206 - 214.
[8] M. Hall and J. K. Senior,
The Groups of Order 2n(n 6), Macmillan, New York, 1964.
[9] M. Hassanzadeh and R. Hatamian, An approach to capable groups and Schur’s theorem,
Bull. Aust. Math. Soc. 92 (2015) 52 - 56.
[10] N. S. Hekster, On the structure of n-isoclinism classes of groups,
J. Pure Appl. Algebra 40 (1986) 63 - 85.
[11] I. M. Isaacs, Derived subgroups and centers of capable groups,
Proc. Amer. Math. Soc. 129 (2001) 2853 - 2859.
[12] M. R. R. Moghaddam and S. Kayvanfar, A new notion derived from varieties of groups,
Algebra Colloq. 4 (1997) 1 - 11.
[13] K. Podosky and B. Szegedy, Bound for the index of the center in capable groups,
Proc. Amer. Math. Soc. 133 (2005) 3441 - 3445.
[14] S. Rashid, N. H. Sarmin, A. Erfanian and N. M. Mohd Ali, On the nonabelian tensor square and capability of groups of order
p2q, Arch. Math. 97 (2011) 299 - 306.
[15] D. J. S. Robinson,
A Course in the Theory of Groups, Springer-Verlag, Berlin, 1982.
[16] D. R. Taunt, On A-groups,
Math. Proc. Cambridge Philos. Soc. 45 (1949) 24 - 42.