Document Type : Research Paper

**Authors**

Department of Pure Mathematics, Ferdowsi University of Mashhad, P. O. Box 1159-91775, Mashhad, Iran

**Abstract**

A famous theorem of Schur states that for a group $G$ finiteness of $G/Z(G)$ implies the finiteness of $G'.$ The converse of Schur's theorem is an interesting problem which has been considered by some authors. Recently, Podoski and Szegedy proved the truth of the converse of Schur's theorem for capable groups. They also established an explicit bound for the index of the center of such groups. This paper is devoted to determine some families of groups among non-capable groups which satisfy the converse of Schur's theorem and at the same time admit the Podoski and Szegedy's bound as the upper bound for the index of their centers.

**Keywords**

**Main Subjects**

R. Baer (1938). Groups with preassigned central and central quotient group. *Trans. Amer. Math. Soc.*. 44, 387-412 J. Burns and G. Ellis (1997). On the nilpotent multipliers of a group. *Math. Z.*. 226, 405-428 N. S. Hekster (1986). On the structure of $n$-isoclinism classes of groups. *J. Pure Appl. Algebra*. 40, 63-85 I. M. Isaacs (2001). Derived subgroups and centers of capable groups. *Proc. Amer. Math. Soc.*. 129, 2853-2859 R. James (1980). The groups of order $p^6$. *Math. Comp.*. 34 (150), 613-637 I. D. Macdonald (1961). Some explicit bounds in groups with finite derived groups. *Proc. London
Math. Soc.*. 3 (11), 23-56 M. R. R. Moghaddam and S. Kayvanfar (1997). A new notion derived from varieties of groups. *Algebra Colloq.*. 4, 1-11 B. H. Neumann (1951). Groups with finite classes of conjugate elements. *Proc. London Math. Soc.*. 3 (1), 178-187 P. Niroomand (2010). The converse of Schur's theorem. *Arch. Math. (Basel)*. 94 (5), 401-403 K. Podoski and B. Szegedy (2002). Bounds in groups with finite Abelian coverings or with finite derived
groups. *J. Group Theory*. 5, 443-452 K. Podoski and B. Szegedy (2005). Bounds for the index of the centre in capable groups. *Proc. Amer. Math. Soc.*. 133 (12), 3441-3445 D. J. S. Robinson (1982). A course in the theory of groups. *Springer-Verlag, New York*. Sh. Shahriari (1987). On normal subgroups of capable groups. *Arch. Math. (Basel)*. 48, 193-198 J. Wiegold (1965). Multiplicators and groups with finite central factor-groups. *Math. Z.*. 89, 345-347 M. Yadav (2010). On finite capable $p$-groups of class 2 with cyclic commutator subgroups. *arXiv:1001.3779v1*.

December 2012

Pages 1-7

**Receive Date:**12 May 2012**Revise Date:**17 July 2012**Accept Date:**17 July 2012**Published Online:**01 December 2012