On indecomposable matrix representations of finite p-groups over commutative local rings
by
Petro Gudivok
Uzhgorod University, Uzhgorod, Ukraine
Coauthors: Igor Chukhraj

Roggenkamp [1] have proved, that if a noetherian domain R of characteristic p > 0 is not field and p divides the order |G| of a finite group G, then there exists infinite number of nonisomorphic indecomposable RG-lattices of finite R-rank less then |G|+1 (RG is a group ring of a group G over a ring R). The problem of finitness of the set of degrees of indecomposable matrix representations of a finite p-group over an arbitrary commutative local ring of characteristic p^s is solved in [2-3].

We are investigating, when a finite p-group has infinite number of nonequivalent indecomposable matrix representations of an arbitrary degree n > 1 over commutative local ring of characteristic p^s. We have received next results.

Theorem 1. Let G be a finite noncyclic p-group (p > 2) and K be an infinite commutative local ring of characteristic p. There exists infinite number of nonequivalent indecomposable matrix K-representations of an arbitrary degree n > 1 of the group G.

Theorem 2. Let G be a finite p-group of the order |G| > 2, K be a commutative local ring of characteristic p^s (s > 0) and Rad K be a jacobson radical of the ring K. If Rad K =/= 0 and K/Rad K is infinite field, then the number of nonequivalent indecomposable matrix K-representations of an arbitrary degree n > 1 of the group G is infinite.

References

[1] K.W. Roggenkamp, Gruppenringe von unendlichem Darstellungstyp, Math. Z. 96 (1967), 393-398.

[2] P.M. Gudivok, V.I. Pogoriljak, On indecomposable representations of finite p-groups over commutative local rings, Dopovidi NAN Ukraini 5 (1996), 7-11.

[3] P.M. Gudivok, V.I. Pogoriljak, On indecomposable matrix representations of finite p-groups over commutative local rings of characteristic p^s, Nauk. visnik Uzhgorod. univ., ser. matem. 4 (1999), 47-53.

Date received: June 29, 2000