Ideals in the modular group ring of a 𝑝-group

Hill, E. T.

doi:10.1090/S0002-9939-1971-0276373-1

Ideals in the modular group ring of a $p$-group
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Proc. Amer. Math. Soc. 28 (1971), 389-390 Request permission

Abstract:

We show that if $G$ has order ${p^n}$ then the group ring has a chain of ${p^n} + 1$ ideals and that the radical powers are canonical in the lattice of ideals. We then prove that if $G$ is abelian, $G$ is determined by the lattice of ideals.

References

W. E. Deskins, Finite Abelian groups with isomorphic group algebras, Duke Math. J. 23 (1956), 35–40. MR 77535, DOI 10.1215/S0012-7094-56-02304-3
E. T. Hill, The annihilator of radical powers in the modular group ring of a $p$-group, Proc. Amer. Math. Soc. 25 (1970), 811–815. MR 262387, DOI 10.1090/S0002-9939-1970-0262387-3
S. A. Jennings, The structure of the group ring of a $p$-group over a modular field, Trans. Amer. Math. Soc. 50 (1941), 175–185. MR 4626, DOI 10.1090/S0002-9947-1941-0004626-6
D. S. Passman, The group algebras of groups of order $p^{4}$ over a modular field, Michigan Math. J. 12 (1965), 405–415. MR 185022, DOI 10.1307/mmj/1028999424