Advertisement

Determination of Certain t-Groups

  • Laxmi Kant MishraEmail author
  • B. K. Sharma
Short Communication
  • 1 Downloads

Abstract

Transiso graph \(\varGamma _d(G)\) is a graph defined for a finite group G and a divisor d of the order of G. Subgroups of G are vertices of \(\varGamma _d(G)\) and two subgroups are said to be connected by an edge if they have a pair of isomorphic normalized right transversals. A group G is called a t-group if graph \(\varGamma _d(G)\) is a complete graph for each divisor d of |G|. Completeness of transiso graphs for some groups like abelian groups, p-groups of order up to \(p^5\) where p is an odd prime, dihedral groups and dicyclic groups etc. are already discussed in the literature. In the present research article, we have discussed the completeness of transiso graphs for 2-groups of the order \(2^5\) and concluded that there are only two non-abelian t-groups (precisely dihedral group and generalized quaternion group) of the order \(2^5\).

Keywords

Transiso graph t-Group Right loop Normalized right transversal 

Mathematics Subject Classification

20N05 05C25 

Notes

References

  1. 1.
    Lal R (1996) Transversals in groups. J Algebra 181(1):70–81MathSciNetCrossRefGoogle Scholar
  2. 2.
    Lal R, Shukla RP (2005) A characterization of Tarski monsters. Indian J Pure Appl Math 36(12):673–678MathSciNetzbMATHGoogle Scholar
  3. 3.
    Kakkar V, Mishra LK (2015) On transiso graph. Asian Eur J Math 8(4):1550070MathSciNetCrossRefGoogle Scholar
  4. 4.
    Mishra LK, Sharma BK (2016) On transiso graphs of groups of order less than 32. J Int Acad Phys Sci 20(2):93–106MathSciNetGoogle Scholar
  5. 5.
    Hall M Jr, Senior JK (1964) The groups of order \(2^n\) (\(n\leqslant 6\)). The Macmillan Co., New YorkzbMATHGoogle Scholar
  6. 6.
    Robinson DJS (1996) A course in the theory of groups. Springer, New YorkCrossRefGoogle Scholar
  7. 7.
    The GAP Group (2015) GAP groups, algorithms and programming, version 4.7.8Google Scholar

Copyright information

© The National Academy of Sciences, India 2019

Authors and Affiliations

  1. 1.Department of MathematicsBhavan’s Mehta MahavidyalayaBharwari, KaushambiIndia
  2. 2.Department of MathematicsUniversity of AllahabadPrayagrajIndia

Personalised recommendations