Document Type: Research Paper


Shahid Rajaee Teacher Training University


H\"{o}lder in 1893 characterized all groups of order $pqr$ where  $p>q>r$ are prime numbers. In this paper,  by using new presentations of these groups, we compute their full automorphism group.


[1] J. E. Adney, T. Yen, Automorphisms of a p-group, Illinois J. Math. 9 (1965) 137-143.
[2] B. E. Earn ley, On nite groups whose group of automorphisms is abelian, PhD thesis, Wayne State University, Detroit, Michigan, 1975. [3] H. Christopher, R. Darren, Automorphisms of nite abelian groups, Amer. Math. Month. 114(10) (2007) 917-923.
[4] M. J. Curran, Semidirect product groups with abelian automorphism groups, J. Austral. Math. Soc. Ser. A 42 (1987) 84-91.
[5] D. Dummit, David, S. Foote, M. Richard, Abstract Algebra (3rd ed.), John Wiley, Sons, 2004.
[6] H. Holder, Die Gruppen der Ordnungen p3; pq2; pqr; p4, Math. Ann. xliii (1893) 371-410.
[7] T. W. Hungerford , Algebra, Springer-Verlag, New York, 1980.
[8] V. K. Jain, P. K. Rai, M. K. Yadav, On Finite p-groups with abelian automorphism group, Inter. J. Alg. Compu. in press.
[9] A. Jamali, Some new non-abelian 2-groups with abelian automorphism groups, J. Group Theory 5 (2002) 53-57.
[10] D. Jonah, M. Konvisser, Some non-abelian p-groups with abelian automorphism groups, Arch. Math. (Basel) 26(1975) 131-133.
[11] E. I. Khukhro, V. D. Mazurov, Finite groups with an automorphism of prime order whose centralizer has small
rank, J. Algebra 301 (2006) 474-492.
[12] G. A. Miller, A non-abelian group whose group of automorphisms is abelian, Messenger Math. 43 (1913) 124-125.
[13] M. Morigi, On p-groups with abelian automorphism group, Rend. Sem. Mat. Univ. Padova 92 (1994) 47-58.
[14] A. Ranum, The group of classes of congruent matrices with application to the group of isomorphisms of any abelian group, Trans. Amer. Math. Soc. 8 (1907) 71{91.
[15] J. Thompson, Automorphisms of solvable groups, J. Algebra 1 (1964) 259-267.
[16] H. Wielandt, Finite Permutation Groups, Academic Press, New York, 1964.