Products Of Element Orders In Finite Groups

Öz

The aim of this note is to investigate the products of element orders in finite groups and to give some properties of such products. Let phi′(G) denote the product of element orders of a finite group G. As an immediate consequence, by using a different approach, we proved that phi′(G) < phi'′(G) where G is a non-cyclic finite group and C is a cyclic group of the same order.

Anahtar Kelimeler

• Finite group
• finite element orders
• product of element orders

Sonlu Grupların Eleman Mertebelerinin Çarpımı

Öz

Bu çalışmanın amacı sonlu grupların eleman mertebelerinin çarpımlarını araştırmak ve bu çarpımların bazı özelliklerini vermektir. Bir sonlu grup G nin eleman mertebelerinin çarpımı phi′(G) olsun. Farklı bir yaklaşım kullanarak aynı mertebeye sahip G devirli olmayan grup ve C devirli grup olmak üzere phi′(G) < phi′(C) olduğu ispatlandı.

Anahtar Kelimeler

• Sonlu grup
• grup eleman mertebeleri
• eleman mertebeleri çarpımı

Kaynakça

1. Amiri, H., Amiri, S.M.J. and Isaacs, I.M. 2009. Sums of element orders in finite groups, Comm. Algebra. Volume 37, 2978-2980.
2. Garonzi, M. and Patassini, M. 2017. Inequality detecting structural properties of a finite group, Comm. Algebra, 45, 677-687.
3. Herzog, M., Longobardi, P. And Maj, M. 2018. An exact upper bound for sums of element orders in non-cyclic finite groups, J. Pure and Applied Algebra, 222 (7), 1628-1642.
4. Isaacs, I.M. (2018). Finite group theory, Graduate Studies in Mathematics, 92, American Mathematical Society, Providence, RI.
5. Mansuroğlu, N. 2018. Sums of element orders in symmetric groups, Afyon Kocatepe University Journal of Science and Engineering, 18, 884-887.
6. Robinson, D.J.S. (1996). A Course in the Theory of Groups, 2nd edition, SpringerVerlag, New York.
7. Tarnauceanu, M. (2018). A note on theproduct of element orders of finite abelian groups, arXiv:1805.09310vl