J. Aust. Math. Soc.  81 (2006), 1-9
The commutator subgroup and Schur multiplier of a pair of finite p-groups

Ali Reza Salemkar
  Faculty of Mathematical Sciences
  Shahid Beheshti University
  Tehran
  Iran
  salemkar@hamoon.usb.ac.ir
Mohammad Reza R. Moghaddam
  Centre of Excelence in
  Analysis on Algebraic Structures
  (President)
  and
  Faculty of Mathematical Sciences
  Ferdowsi University of Mashhad
  Iran
  moghadam@math.um.ac.ir
and
Farshid Saeedi
  Department of Mathematics
  Azad University of Mashhad
  Iran
  saeedi@mshdiau.ac.ir


Abstract
Let $(M,G)$ be a pair of groups, in which $M$ is a normal subgroup of $G$ such that $G/M$ and $M/Z(M,G)$ are of orders $p^m$ and $p^n$, respectively. In 1998, Ellis proved that the commutator subgroup $[M,G]$ has order at most $p^{n(n+2m-1)/2}$.

In the present paper by assuming $|[M,G]|=p^{n(n+2m-1)/2}$, we determine the pair $(M,G)$. An upper bound is obtained for the Schur multiplier of the pair $(M,G)$, which generalizes the work of Green (1956).
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