Orbits classifying extensions of prime power order groups

Document Type : Research Paper


1 Matematika Saila, Euskal Herriko Unibertsitatearen Zientzia eta Teknologia Fakultatea, Posta-kutxa 644, 48080 Bilbo, Spain

2 Dipartimento di Matematica, Università di Trento, via Sommarive 14, 38123 Povo (Trento), Italy



The strong isomorphism classes of extensions of finite groups are parametrized by orbits of a prescribed action on the second cohomology group. We study these orbits in the case of extensions of a finite abelian $p$-group by a cyclic factor of order $p$. As an application, we compute the number and sizes of these orbits when the initial $p$-group is generated by at most $3$ elements.


Main Subjects

[1] H. U. Besche and B. Eick, Construction of finite groups, J. Symbolic Comput., 27 (1999) 387–404.
[2] S. R. Blackburn, Groups of prime power order with derived subgroup of prime order, J. Algebra, 219 (1999) 625–657.
[3] K. S. Brown, Cohomology of groups, Graduate Texts in Mathematics, 87, Springer–Verlag, New York–Berlin, 1982.
[4] H. Dietrich and B. Eick, On the groups of cube-free order, J. Algebra, 292 (2005) 122–137.
[5] H. Dietrich, B. Eick and D. Feichtenschlager, Investigating p-groups by coclass with GAP, Computational group theory and the theory of groups, Contemp. Math., 470, Amer. Math. Soc., Providence, RI, 2008 45–61.
[6] B. Eick and E. A. O’Brien, Enumerating p-groups, J. Austral. Math. Soc. Ser. A, 67 (1999) 191–205.
[7] L. Evens, The cohomology of groups, Oxford Mathematical Monographs. Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1991.
[8] H. Fitting, Beiträge zur Theorie der Gruppen endlicher Ordnung, Jahresber. Dtsch. Math.-Ver., 48 (1938) 77–141.
[9] J. A. Grochow and Y. Qiao, Algorithms for group isomorphism via group extensions and cohomology, SIAM J.
Comput., 46 (2017) 1153–1216.
[10] J. R. Harper, Secondary cohomology operations, Graduate Studies in Mathematics, 49, American Mathematical Soci-ety, Providence, RI, 2002.
[11] R. Laue, Zur Konstruktion und Klassifikation endlicher auflösbarer Gruppen, Bayreuth. Math. Schr., No. 9 (1982).
[12] J. McCleary, A user’s guide to spectral sequences, 2nd ed., Cambridge University Press, 2001.
[13] D. V. Millionshchikov and R. Khimenes, Geometry of central extensions of nilpotent Lie algebras, Tr. Mat. Inst.
Steklova, 305 (2019) 225–249.
[14] S. A. Morris, Pontryagin duality and the structure of locally compact abelian groups, London Mathematical Society Lecture Note Series, No. 29, Cambridge University Press, Cambridge-New York-Melbourne, 1977.
[15] M. Michalek and B. Sturmfels, Invitation to nonlinear algebra, Graduate Studies in Mathematics, 211, American
Mathematical Society, Providence, RI, 2021.
[16] E. A. O’Brien, The p-group generation algorithm, Computational group theory, Part 1, J. Symbolic Comput., 9 (1990) 677–698.
[17] D. J. S. Robinson, Applications of cohomology to the theory of groups, In Campbell, C. M. Groups - St. Andrews 1981, LMS Lecture Note Series, 71, Cambridge, Cambridge University Press, 1981.
[18] I. R. Shafarevich, Basic algebraic geometry 1, Varieties in projective space. Third edition, Translated from the 2007 third Russian edition, Springer, Heidelberg, 2013.
[19] A. C. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, 38, Cambridge University Press, Cambridge, 1994.