Jafari, S. (2017). Characterization of finite $p$-groups by the order of their Schur multipliers ($t(G)=7$). Bulletin of the Iranian Mathematical Society, 43(7), 2567-2576.

S.H. Jafari. "Characterization of finite $p$-groups by the order of their Schur multipliers ($t(G)=7$)". Bulletin of the Iranian Mathematical Society, 43, 7, 2017, 2567-2576.

Jafari, S. (2017). 'Characterization of finite $p$-groups by the order of their Schur multipliers ($t(G)=7$)', Bulletin of the Iranian Mathematical Society, 43(7), pp. 2567-2576.

Jafari, S. Characterization of finite $p$-groups by the order of their Schur multipliers ($t(G)=7$). Bulletin of the Iranian Mathematical Society, 2017; 43(7): 2567-2576.

Characterization of finite $p$-groups by the order of their Schur multipliers ($t(G)=7$)

^{}Department of Mathematics, Mashhad branch, Islamic Azad University, Mashhad, Iran

Receive Date: 12 April 2017,
Revise Date: 09 December 2017,
Accept Date: 26 December 2017

Abstract

Let $G$ be a finite $p$-group of order $p^n$ and $|{\mathcal M}(G)|=p^{\frac{1}{2}n(n-1)-t(G)}$, where ${\mathcal M}(G)$ is the Schur multiplier of $G$ and $t(G)$ is a nonnegative integer. The classification of such groups $G$ is already known for $t(G)\leq 6$. This paper extends the classification to $t(G)=7$.

Ya.G. Berkovich, On the order of the commutator subgroups and the Schur multiplier of a finite p-group, J. Algebra 144 (1991) 269-272.

R. Brown, D.L. Johnson and E.F. Robertson, Some computations of nonabelian tensor products of groups, J. Algebra 111 (1987) 177--202.

R. Brown and J.-L. Loday, Van Kampen theorems for diagrams of spaces, Topology 26 (1987) 311-335.

S. Cicalo, W.A. De Graaf and M. Vaughan-Lee, An effective version of the Lazard correspondence, J. Algebra 352 (2012) 430-450.

B. Eick, M. Horn and S. Zandi, Schur multipliers and the Lazard correspondence, Arch. Math. 99 (2012) 217--226.

G. Ellis, On the Schur multipliers of p-groups, Comm. Algebra 27(9) (1999) 4173--4177.

G. Ellis and J. Wiegold, A bound on the Schur multiplier of a prime-power group, Bull. Austral. Math. Soc. 60(2) (1999) 191--196.

The GAP Group, GAP-Groups, Algorithms and Programming, Version 4.7.6, 2014 (http://www.gap-system.org/).

J.A. Green, On the number of automorphisms of a finite group, Proc. Royal Soc. London Ser. A. 237 (1956) 574--581.

P. Hardy and E. Stitzinger, On characterizing nilpotent Lie algebras by their multipliers, t(L) = 3; 4; 5; 6, Comm. Algebra 26(11) (1998) 3527-3539.

S.H. Jafari, F. Saeedi and E. Khamseh, Characterization of finite p-groups by their nonabelian tensor square, Comm. Algebra 41(5) (2013) 1954--1963.

S.H. Jafari, Categorizing finite p-groups by the order of their nonabelian tensor squares, J. Algebra Appl. 15 (2016), no. 5, 1650095, 13 pages.

S.H. Jafari, Characterization of finite p-groups by the order of their Schur multipliers (t = 6), Math. Rep. (Bucur.) 18(68) (2016), no. 4, 535--543.

R. James, The groups of order p6 (p an odd prime), Math. Comp. 34 (1980) 613--637.

M.R. Jones, Some inequalities for the multiplicator of a finite group, Proc. Amer. Math. Soc. 39 (1973) 450--456.

G. Karpilovsky, The Schur Multiplier, London Math. Soc. Monogr. New Series 2, Clarendon Press, New York, 1987.

E. Khamseh, M.R.R. Moghaddam anf F. Saeedi, Characterization of finite p-groups by their Schur multiplier, J. Algebra Appl. 12 (2013), no. 5, 1250035, 9 pages.

P. Niroomand, On the order of Schur multiplier of non-abelian p-groups, J. Algebra 322 (2009) 4479--4482.

P. Niroomand, Characterizing finite p-groups by their Schur multipliers, C. R. Math. Acad. Sci. Paris 350 (2012) 867--870.

P. Niroomand, Characterizing finite p-groups by their Schur multipliers, t(G) = 5, Math. Rep. (Bucur.) 17(67) (2015), no. 2, 249--254.

E.A. O'Brien and M.R. Vaughan-Lee, The groups with order p7 for odd prime p, J. Algebra 292 (2005), no. 1, 243--258.

I. Schur, Uber die Darstellungen der endlichen Gruppen durch gebrochene lineare Sub-stitutionen, J. Reine Angew. Math. 127 (1904) 20--50.

M. Vaughan-Lee, Groups of order p8 and exponent p, Int. J. Group Theory 4(4) (2015) 25--42.

X. Zhou, On the order of Schur multipliers of finite p-groups, Comm. Algebra 22 (1994), no. 1, 1--8.