在這篇論文裡，我們介紹一些關於幾個特定有限群的共軛類乘積的結果。我們發現了某些性質，並提出幾個猜想。除此之外，我們也介紹關於共軛類大小的一些相關性問題。

In this thesis, we introduce some results for product of two conjugacy classes in some finite groups. We also find some properties and present some conjectures. In addition, we introduce some related problems about the size of conjugacy classes.

[1] E. Adan-Bante, Homogeneous products of conjugacy classes, Arch. Math. (Basel) 86 (2006), no. 4, 289-294.
[2] E. Adan-Bante and J. M. Harris, On conjugacy classes of $GL(n,q)$ and $SL(n,q)$, preprint.
[3] E. Adan-Bante and J. M. Harris, On conjugacy classes of $SL(2,q)$, preprint.
[4] E. Adan-Bante, J. M. Harris, and H. Verrill, Products of conjugacy classes of the alternating group, preprint.
[5] E. Adan-Bante, M. Loukaki, and A. Moret o, Homogeneous products of characters, J. Algebra 274 (2004), no. 2, 587-593.
[6] E. Adan-Bante and H. Verrill, Symmetric groups and conjugacy classes, J. Group Theory 11 (2008), no. 3, 371-379.
[7] N. Ahanjideh, On Thompson's conjecture for some nite simple groups, J. Algebra 344 (2011), 205-228.
[8] A. G. Aleksandrov and K. A. Komissarcik, Simple groups with a small number of conjugacy classes, In Algorithmic Studies in Combinatorics (Russian), Nauka, Moscow 187 (1978), 162-172.
[9] Z. Arad and M. Herzog, Products of conjugacy classes in groups, Lecture Notes in Mathematics, vol. 1112, Springer-Verlag, 1985.
[10] Z. Arad, M. Muzychuk, and A. Oliver, On groups with conjugacy classes of distinct sizes, J. Algebra 280 (2004), no. 2, 537-576.
[11] R. Brauer, Representations of nite groups, Lectures on Modern Mathematics, pp. 133-175, Wiley, 1963.
[12] J. L. Brenner, M. Randall, and J. Riddell, Covering theorems for FINASIGS, I, Colloq. Math. 32 (1974), 39-48.
[13] W. Burnside, Theory of groups of nite order, 2 ed., Dover Publications, 1911.
[14] D. Chillag and M. Herzog, Finite groups with extremal conditions on sizes of conjugacy classes and on degrees of irreducible characters, Groups St. Andrews 1 (2005), 269- 273.
[15] D. S. Dummit and R. M. Foote, Abstract algebra, 3 ed., John Wiley & Sons, Inc., 2004.
[16] E. W. Ellers and N. Gordeev, On the conjectures of J. Thompson and O. Ore, Trans. Amer. Math. Soc (1998), no. 9, 3657-3671.
[17] GAP, Groups, Algorithms, Programming - a System for Computational Discrete Algebra, 2008, Version 4.4.12.
[18] A. L. Gilotti, Sui gruppi niti in cui classi distinte di elementi coniugati hanno diversa cardinalit a (Italian), Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 58 (1975), no. 4, 501-507.
[19] R. M. Guralnick and G. Malle, Products of conjugacy classes and xed point spaces, J. Amer. Math. Soc. 25 (2012), no. 1, 77-121.
[20] M. Hayashi, On a generalization of F. M. Markel's Theorem, Hokkaido Math. J. 4 (1975), 278-280.
[21] M. Herzog and J. Schonheim, On groups of odd order with exactly two non-central conjugacy classes of each size, Arch. Math. (Basel) 86 (2006), no. 1, 7-10.
[22] G. James and M. Liebeck, Representations and characters of groups, 2 ed., Cambridge University Press, 2001.
[23] L. C. Kappe and R. F. Morse, On commutators in groups, Groups St. Andrews 2005 2, 531-558.
[24] R. Knorr, W. Lempken, and B. Thielcke, The S3-conjecture for solvable groups, Israel J. Math. 91 (1995), no. 1-3, 61-76.
[25] L. F. Kosvintsev, Over the theory of groups with properties given over the centralizers of involutions, 1974, Sverdlovsk (Ural.), Summary thesis Doct.
[26] E. Landau,  Uber die Klassenzahl der binaren quadratischen Formen von negativer Discriminante (German), Math. Ann. 56 (1903), no. 4, 671-676.
[27] M. W. Liebeck, E. A. O'Brien, A. Shalev, and P. H. Tiep, The Ore conjecture, J. Eur. Math. Soc. (JEMS) 12 (2010), no. 4, 939-1008.
[28] A. V. L opez and J. V. L opez, Classi cation of nite nilpotent groups according to the number of conjugacy classes and of minimal normal subgroups (Spanish), Proceedings of the Eighth Portuguese-Spanish Conference on Mathematics, vol. I, Univ. Coimbra, Coimbra, 1981, pp. 245-252.
[29] A. V. L opez and J. V. L opez, Classi cation of nite groups according to the number of conjugacy classes, Israel J. Math. 51 (1985), no. 4, 305-338.
[30] F. M. Markel, Conjugacy class problem for nite groups, Ph.D. thesis, University of Toronto (Canada), 1973.
[31] F. M. Markel, Groups with many conjugate elements, J. Algebra 26 (1973), 69-74.
[32] G. A. Miller, Groups involving only a small number of sets of conjugate operators, Arch. Math. and Phys. 17 (1910), 199-204.
[33] V. A. Odincov and A. I. Starostin, Finite groups with 9 classes of conjugate elements (Russian), Ural. Gos. Univ. Mat. Zap. 10, Issled Sovremen, Algebre 152 (1976), no. 1, 114-134.
[34] O. Ore, Some remarks on commutators, Proc. Amer. Math. Soc. 2 (1951), 307-314.
[35] J. Poland, Finite groups with a given number of conjugate classes, Canad. J. Math. 20 (1968), 456-464.
[36] J. J. Rotman, An introduction to the theory of groups, 4 ed., Graduate Texts in Mathematics, no. 148, Springer-Verlag, 1995.
[37] D. T. Sigley, Groups involving ve complete sets of non-invariant conjugate operators, Duke Math. J. 1 (1935), no. 4, 477-479.
[38] M. B. Ward, Finite groups in which no two distinct conjugacy classes have the same order, Arch. Math. (Basel) 54 (1990), no. 2, 111-116.
[39] J. P. Zhang, Finite groups with many conjugate elements, J. Algebra 170 (1994), no. 2, 608-624.