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 Please use this identifier to cite or link to this item: http://hdl.handle.net/1974/5258

Title: Invariants of Modular Two-Row Groups
Authors: Wu, YINGLIN

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Keywords: Invariants
Modular
Two-Row Group
Complete Intersection
Nakajima p-Group
Issue Date: 2009
Abstract: It is known that the ring of invariants of any two-row group is Cohen-Macaulay. This result inspired the conjecture that the ring of invariants of any two-row group is a complete intersection. In this thesis, we study this conjecture in the case where the ground field is the prime field $\mathbb{F}_p$. We prove that all Abelian reflection two-row $p$-groups have complete intersection invariant rings. We show that all two-row groups with \textit{non-normal} Sylow $p$-subgroups have polynomial invariant rings. We also show that reflection two-row groups with \textit{normal} reflection Sylow $p$-subgroups have polynomial invariant rings. As an interesting application of a theorem of Nakajima about hypersurface invariant rings, we rework a classical result which says that the invariant rings of subgroups of $\text{SL}(2,\,p)$ are all hypersurfaces. In addition, we obtain a result that characterizes Nakajima $p$-groups in characteristic $p$, namely, if the invariant ring is generated by norms, then the group is a Nakajima $p$-group.