Automorphism Group of Dihedral Groups With Perfect Order Subsets

Authors

  • Vinod S. Department of Mathematics, Government College for Women, Thiruvananthapuram, Kerala
  • Biju G. S. Department of Mathematics, College of Engineering, Thiruvananthapuram 695016, Kerala

DOI:

https://doi.org/10.26713/cma.v12i2.1513

Keywords:

Dihedral group, Order classes, Automorphism, Conjugacy classes

Abstract

Let \(G\) be a finite group. The set of all possible such orders joint with the number of elements that each order referred to, is called the order classes of \(G\). The order subset of \(G\) determined by \(x\in G\) is the set of elements in \(G\) with the same order as \(x\). A group is said to have perfect order subsets (POS-group) if the cardinality of each order subset divides the group order. In this paper, we compute the order classes of the automorphism group of Dihedral group. Also, we construct a class of POS groups from the automorphism group of the Dihedral group which will serve the solution to the Perfect Order Subset Conjecture.

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References

B. Al-Hasanat, A. Ahmad and H. Sulaiman, Order classes of dihedral groups, AIP Conference Proceedings 1605(1) (2014), 551 – 556, DOI: 10.1063/1.4887648.

S.R. Cavior, The subgroups of the dihedral group, Mathematics Magazine 48(2) (1975), p. 107, DOI: 10.1080/0025570X.1975.11976454.

K. Conrad, Dihedral groups II, URL: https://kconrad.math.uconn.edu/blurbs/grouptheory/dihedral2.pdf (accessed: December 2020).

K. Conrad, Dihedral groups, URL: http://www.math.uconn.edu/~kconrad/blurbs/grouptheory/dihedral.pdf (accessed: December 2020).

A.K. Das, On finite groups having perfect order subsets, International Journal of Algebra 3(29) (2009), 629 – 637.

X. Du and W. Shi, Finite groups with conjugacy classes number one greater than its same order classes number, Communications in Algebra 34 (2006), 1345 – 1359, DOI: 10.1080/00927870500454638.

D.S. Dummit and R.M. Foote, Abstract Algebra, Wiley Hoboken (2003).

C.E. Finch and L. Jones, A curious connection between Fermat numbers and finite groups, American Mathematical Monthly 109(6) (2002), 517 – 524, DOI: 10.1080/00029890.2002.11919881.

C.E. Finch and L. Jones, Nonabelian groups with perfect order subsets, JP Journal of Algebra, Number Theory and Application 3(1) (2003), 13 – 26.

R.M. Foote and B.M. Reist, The perfect order subset conjecture for simple groups, Journal of Algebra 391 (2013), 1 – 21, DOI: 10.1016/j.jalgebra.2013.05.029.

J.A. Gallian, Contemporary Abstract Algebra, D.C. Heath and Company (1994).

L. Jones and K. Toppin, On three questions concerning groups with perfect order subsets, Involve, A Journal of Mathematics 4(3) (2012), 251 – 261, DOI: 10.2140/involve.2011.4.251.

S. Libera and P. Tlucek, Some perfect order subset groups, Pi Mu Epsilon Journal 11(9) (2003), 495 – 498, https://www.jstor.org/stable/24340521.

R. Shen, A note on finite groups having perfect order subsets, International Journal of Algebra 4(13-16) (2010), 643 – 646.

N.T. Tuan and B.X. Hai, On perfect order subsets in finite groups, International Journal of Algebra 4(21-24) (2010), 1021 – 1029.

A.V. Vasil'ev, M.A. Grechkoseeva and V.D. Mazuro, Characterization of the finite simple groups by spectrum and order, Algebra and Logic 48(6) (2009), 385 – 409, https://link.springer.com/content/pdf/10.1007/s10469-009-9074-9.

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Published

30-06-2021
CITATION

How to Cite

S., V., & S., B. G. (2021). Automorphism Group of Dihedral Groups With Perfect Order Subsets. Communications in Mathematics and Applications, 12(2), 263–271. https://doi.org/10.26713/cma.v12i2.1513

Issue

Vol. 12 No. 2 (2021)

Section

Research Article

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