## Invariants of Modular Two-Row Groups

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##### Date

2009-10-06T19:50:09Z

##### Authors

Wu, Yinglin

##### Keyword

Invariants , Modular , Two-Row Group , Complete Intersection , Nakajima p-Group

##### 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.