Geodesic
Decomposition of Grothendieck group of stable categories of finite groups
Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 6, Tome 265 (1999), pp. 163-168
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Let $G$ be a finite group with a normal Sylow $p$-subgroup $H$ such that the corresponding quotient is abelian. We prove that the Grothendieck group of the stable category of $G$ (over an algebraically closed field with characteristic $p$) contains a cyclic direct summand of order $|H|$. 
@article{ZNSL_1999_265_a10,
     author = {A. I. Generalov and I. O. Isametdinov},
     title = {Decomposition of {Grothendieck} group of stable categories of finite groups},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {163--168},
     year = {1999},
     volume = {265},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1999_265_a10/}
}
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A. I. Generalov; I. O. Isametdinov. Decomposition of Grothendieck group of stable categories of finite groups. Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 6, Tome 265 (1999), pp. 163-168. http://geodesic.mathdoc.fr/item/ZNSL_1999_265_a10/