A Simple Classification of Finite Groups of Order p2q2

Document Type : Original Scientific Paper


Department of Mathematics, Faculty of Science, Shahid Rajaee Teacher Training University


‎Suppose G is a group of order p2q2 where p>q are prime numbers and suppose P and Q are Sylow p-subgroups and Sylow q-subgroups of G, ‎respectively‎. ‎In this paper‎, ‎we show that up to isomorphism‎, ‎there are four groups of order p2q2 when Q and P are cyclic‎, ‎three groups when Q is a cyclic and P is an elementary ablian group‎, ‎p2+3p/2+7 groups when Q is an elementary ablian group and P is a cyclic group and finally‎, ‎p‎ + ‎5 groups when both Q and P are elementary abelian groups.‎


Main Subjects

1. ‎W‎. ‎Burnside‎, ‎Theory of Groups of Finite Order‎, ‎2d ed‎. ‎Dover Publications‎, ‎Inc.‎, ‎New York‎, ‎1955‎.
2. ‎D‎. ‎S‎. ‎Dummit‎, ‎R‎. ‎M‎. ‎Foote‎, ‎Abstract Algebra‎, ‎Third Edition‎, ‎John Wiley & Sons‎, ‎Inc.‎, ‎Hoboken‎, ‎NJ‎, ‎2004‎.
3. ‎M‎. ‎Hall‎, ‎On the number of Sylow subgroups in a finite group‎, ‎J‎. ‎Algebra 7 (1967) 363-371‎.
4. ‎B‎. ‎Huppert‎, ‎Endliche Gruppen I.‎, ‎Springer-Verlag‎, ‎Berlin‎, ‎1967‎.
5. C‎. ‎Leedham-Green‎, ‎S‎. ‎McKay‎, The Structure of Groups of Prime Power Order‎, ‎London Mathematical Society Monographs‎. ‎New Series‎, ‎27‎. ‎Oxford Science Publications‎. ‎Oxford University Press‎, ‎Oxford‎, ‎2002‎.
6. ‎L‎. ‎Pyber‎, ‎Enumerating finite groups of given order‎, ‎Ann‎. ‎of Math. 137 (1993) 203-220‎.
7. D‎. ‎J‎. ‎S‎. ‎Robinson‎, ‎A Course in the Theory of Groups, ‎Springer-Verlag‎, ‎New York‎, ‎1982‎.
8. ‎J‎. ‎J‎. ‎Rotman‎, ‎An Introduction to the Theory of Groups, ‎Fourth edition‎, ‎Graduate Texts in Mathematics‎, ‎148‎, Springer-Verlag‎, ‎New York‎, ‎1995‎.
9. ‎R‎. ‎Solomon‎, ‎A brief history of the classification of the finite simple groups‎, Bull‎. ‎Amer‎. ‎Math‎. ‎Soc‎. ‎(N.S.) 38 (2001) 315-352‎.
10. ‎M‎. ‎Suzuki‎, ‎Group Theory I., ‎Springer-Verlag‎, ‎Berlin-New York‎, ‎1982‎.
11. ‎M‎. ‎Suzuki‎, ‎Group Theory II.‎, ‎Springer-Verlag‎, ‎Berlin-New York‎, ‎1986‎.