A note on finite C-tidy groups

Document Type: Research Paper

Author

North-eastern Hill University

Abstract

Let $G$ be a group and $x \in G$‎. ‎The cyclicizer of $x$ is defined to be the subset $Cyc(x)=\lbrace y \in G \mid \langle x‎, ‎y\rangle \; {\rm is \; cyclic} \rbrace$‎. ‎$G$ is said to be a tidy group if $Cyc(x)$ is a subgroup for all $x \in G$‎. ‎We call $G$ to be a C-tidy group if $Cyc(x)$ is a cyclic subgroup for all $x \in G \setminus K(G)$‎, ‎where $K(G)$ is the intersection of all the cyclicizers in $G$‎. ‎In this note‎, ‎we classify finite C-tidy groups with $K(G)=\lbrace 1 \rbrace$‎.

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A. Abdollahi, S. Akbari and H. R. Maimani (2006). Non-commuting graph of a group. J. Algebra. 298, 468-492
A. Abdollahi and A. M. Hassanabadi (2007). Non-cyclic graph of a group. Comm. Algebra. 35, 2057-2081
A. Abdollahi and A. M. Hassanabadi (2009). Non-cyclic graph associated with a group. J. Algebra Appl.. 8, 243-257
A. Abdollahi, A. Azad, A. M. Hassanabadi and M. Zarrin (2010). On the clique numbers of non-commuting graphs of certain groups. Algebra Colloq.. 14 (4), 611-620
Y. G. Berkovich and E. M. Zhmu$acute{d}$ (1998). Characters of Finite groups, Part 1. Transl. Math. Monographs, Amer. Math. Soc., Providence. RI. 172
A. Erfanian and D. G. M. Farrokhi (2008). On some classes of tidy groups. Algebras Groups Geom.. 25 (1), 109-114
M. J. Evans (1993). $T$-systems of certain finite simple groups. Math. Proc. Cambridge Philos. Soc.. 113, 9-22
D. Hughes and J. G. Thompson (1963). The $H_p$-problem and the structure of the $H_p$-groups. Pacific J. Math.. 9, 1097-1102
I. M. Isaacs (1973). Equally partitioned groups. Pacific J. Math.. 49, 109-116
I. M. Isaacs (1976). Character theory of finite groups. Academic Press, Inc., London, New York.
O. H. Kegel (1960/1961). Die nilpotenz der $H_p$-gruppen. Math. Z.. 71, 373-376
O. H. Kegel (1961). Nicht-einfache Partitionen endlicher gruppen. Arch. Math. (Basel). 12, 170-175
G. A. Miller (1906/1907). Groups in which all the operators are contained in a series of subgroups such that any two have only identity in common. Bull. Amer. Math. Soc.. 17, 446-449
L. Mousavi (2011). $n$-Cyclicizer Groups. Bull. Iranian Math. Soc.. 37 (1), 161-170
K. O'Bryant, D. Patrick, L. Smithline and E. Wepsic (1992). Some facts about cycels and tidy groups. Rose-Hulman Institute of Technology, Indiana, USA, Technical report, MS-TR 92-04.
D. Patrick, E. Wepsic (1991). Cyclicizers, centralizers and normalizers. Rose-Hulman Institute of Technology, Indiana, U.S.A, Technical report, MS-TR 91-05.
R. H. Schulz (1985). Transversal designs and partitions associated with Frobenius groups. J. Reine Angew. Math.. 355, 153-162
M. Suzuki (1961). On a finite group with a partition. Arch. Math. (Basel). 12, 241-257
G. Zappa (2003). Partitions and other coverings of finite groups. Illinois J. Math.. 47 (1/2), 571-580