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Proceedings of the American Mathematical Society

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On double centralizer subgroups of some finite $ p$-groups

Author: Ying Cheng
Journal: Proc. Amer. Math. Soc. 86 (1982), 205-208
MSC: Primary 20D15
MathSciNet review: 667273
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Abstract: Let $ A$ be a maximal abelian normal subgroup of a finite $ p$-group $ G(p > 2)$ such that $ [G,A]$ is cyclic. Then (i) $ {C_G}({C_G}(D)) = D$ and $ [G:{C_G}(D)] = [D:Z(G)]$ for every $ Z(G) \leqslant D \leqslant G$; (ii) $ [G:Z(G)] = {[G,A]^2}$ and every faithful absolutely irreducible representation of $ G$ is of degree $ [G:A]$. The case $ p = 2$ will also be mentioned.

References [Enhancements On Off] (What's this?)

  • [1] Y. Cheng, On finite $ p$-groups with cyclic commutator subgroup, Arch. Math. (to appear).
  • [2] Frank DeMeyer and Edward Ingraham, Separable algebras over commutative rings, Lecture Notes in Mathematics, Vol. 181, Springer-Verlag, Berlin-New York, 1971. MR 0280479
  • [3] Daniel Gorenstein, Finite groups, Harper & Row, Publishers, New York-London, 1968. MR 0231903
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  • [6] Thomas J. Laffey, Centralizers of elementary abelian subgroups in finite 𝑝-groups, J. Algebra 51 (1978), no. 1, 88–96. MR 0472997

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Keywords: Commutator subgroup, Azumaya algebras
Article copyright: © Copyright 1982 American Mathematical Society