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Conjugate $p$-elements of Full Support that Generate the Wreath Product $C_{p}\wr C_{p}$ | ||
| International Journal of Group Theory | ||
| مقاله 2، دوره 5، شماره 3، آذر 2016، صفحه 9-35 اصل مقاله (289.15 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22108/ijgt.2016.7806 | ||
| نویسنده | ||
| David Ward* | ||
| University of Manchester | ||
| چکیده | ||
| For a symmetric group $G:=sym{n}$ and a conjugacy class $\mathcal{X}$ of involutions in $G$, it is known that if the class of involutions does not have a unique fixed point, then - with a few small exceptions - given two elements $a,x\in \mathcal{X}$, either $<a,x>$ is isomorphic to the dihedral group $D_{8}$, or there is a further element $y\in \mathcal{X}$ such that $<a,y> ≅ <x,y> ≅ D_8$ (P. Rowley and D. Ward, On $\pi$-Product Involution Graphs in Symmetric Groups. MIMS ePrint, 2014). One natural generalisation of this to $p$-elements is to consider when two conjugate $p$-elements generate a wreath product of two cyclic groups of order $p$. In this paper we give necessary and sufficient conditions for this in the case that our $p$-elements have full support. These conditions relate to given matrices that are of circulant or permutation type, and corresponding polynomials that represent these matrices. We also consider the case that the elements do not have full support, and see why generalising our results to such elements would not be a natural generalisation. | ||
| کلیدواژه‌ها | ||
| circulant matrix؛ cyclic group؛ wreath product | ||
| مراجع | ||
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[1] J. Ballantyne, On lo cal fusion graphs of nite Coxeter groups, J. Group Theory, 16 (2013) 595-617.
[2] J. Ballantyne, N. Greer and P. Rowley, Lo cal Fusion Graphs for Symmetric Groups, J. Group Theory, 16 (2013) 35-49.
[3] C. Bates, D. Bundy, S. Hart and P. Rowley, A Note on Commuting Graphs for Symmetric Groups, Electron. J. Combin., 16 (2009) pp. 13. [4] C. Bates, D. Bundy, S. Perkins and P. Rowley, Commuting involution graphs for symmetric groups, J. Algebra, 266 (2003) 133-153.
[5] C. Bates, D. Bundy, S. Perkins and P. Rowley, Commuting involution graphs for nite Coxeter groups, J. Group Theory, 6 (2003) 461-476.
[6] C. Bates, D. Bundy, S. Perkins and P. Rowley, Commuting Involution Graphs in Sp ecial Linear Groups, Comm. Algebra, 32 (2004) 4179-4196.
[7] J. R. Britnell and N. Gill, Perfect commuting graphs, ArXiv e-prints, 1309.2237, 2013.
[8] K. S. Brown, Euler Characteristics of Groups: The p -Fractional Part, Invent. Math., 29 (1975) 1-5.
[9] K. S. Brown, High dimensional cohomology of discrete groups, Proc. Nat. Acad. Sci. U. S. A., 73 (1976) 1795-1797.
[10] Y. Drozd and R. V. Skuratovskii, Generators and Relations for Wreath Pro ducts, Ukrainian Math. J., 60 (2008) 1168-1171. [11] A. Everett, Commuting Involution Graphs for 3-Dimensional Unitary Groups, Electron. J. Combin., 18 (2011) 11 pp. [12] I. Kra and S. Simanca, On circulant matrices, Notices Amer. Math. Soc., 59 (2012) 368-377. [13] G. L. Morgan and C. W. Parker, The diameter of the commuting graph of a nite group with trivial centre, J. Algebra, 393 (2013) 41-59. [14] C. W. Parker, The commuting graph of a soluble group, Bul l. Lond. Math. Soc., 45 (2013) 839-848. [15] S. Perkins, Commuting involution graphs for An , Arch. Math. (Basel), 86 (2006) 16-25. [16] D. Quillen, Homotopy Prop erties of the Poset of Nontrivial p-Subgroups of a Group, Adv. in Math., 28 (1978) 101-128. [17] M. A. Ronan, Duality for Presheaves on Chamb er Systems and a Related Chain Complex, J. Algebra, 121 (1989) 263-274. [18] M. A. Ronan and S. D. Smith, Sheaves on Buildings and Mo dular Representations of Chevalley Groups, J. Algebra, 96 (1985) 319-346. [19] M. A. Ronan and S. D. Smith, Universal Presheaves on Group Geometries and Mo dular Representations, J. Algebra, 102 (1986) 133-154. [20] M. A. Ronan and S. D. Smith, Computation of 2-mo dular sheaves and representations for L4 (2); A7; 3 S6; and M24, Comm. Algebra, 17 (1989) 1199-1237. [21] P. Rowley and D. Ward, On -Pro duct Involution Graphs in Symmetric Groups, MIMS ePrint , 2014. [22] M. R. Salarian, Characterizing some small simple groups by their commuting involution graphs, Southeast Asian Bul l. Math., 35 (2011) 467-474. [23] J. Shareshian, Hyp ergraph matching complexes and Quillen complexes of symmetric groups, J. Combin. Theory Ser. A, 106 (2004) 299-314. | ||
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