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**Bewertete p-Gruppen und ein Satz von Szele.**
*(German)*
Zbl 0575.20048

It is a result of Szele that a \(p^ n\)-bounded subgroup B of an abelian p-group G is a direct summand if \(B\cap p^ nG=0\). This result is generalized to valuated groups. More precisely, if A is a finite valuated p-group then conditions are found which force any monomorphism \(\alpha\) : \(\oplus_{I}A\to G\) to split where G is a p-local valuated group. As replacement for the classical subgroup \(p^ nG\) the radical \(R_ A(G)=\cap \{Ker(f):\) \(f\in Morph(G,A)\}\) is used. The two generalizations of Szele’s theorem can now be stated.

Corollary 4.3: Let A be a finite, indecomposable, valuated p-group and G a p-valuated group which possesses a morphism \(\phi\) on G to a group \(\oplus_{J}A\) with \(Ker(\phi)=R_ A(G)\). Then every monomorphism \(\alpha\) : \(\oplus_{I}A\to G\) with \(Im(\alpha)\cap R_ A(G)=0\) splits.

The morphism \(\phi\) always exists for simply presented p-groups A and G. Theorem 6.8: The following statements are equivalent for a simply presented, finite, valuated p-group A.

(a) \(A=\oplus^{n}_{i=1}\oplus_{m_ i}S(X_ i)\) where \(m_ i\) is finite, the \(X_ i\) are finite, irretractable, incomparable, valuated p- trees and S(X) denotes the simply presented valuated p-group with p-basis X.

(b) If P is a direct summand of \(\oplus_{I}A\) then every monomorphism \(\alpha\) on P into a simply presented valuated p-group G splits if and only if \(\alpha (P)\cap R_ A(G)=0.\)

An important tool in the proofs is a category equivalence between the category of direct sums \(\oplus_{I}A\) of a finite valuated p-group A and the category of free Morph(A,A)-modules. Furthermore, a subgroup \(G(T^*)\) of the valuated group G, defined for any valuated p-tree T, is discussed at some length as a possible replacement of the classical subgroup \(p^ nG\). However, when both A and G are simply presented valuated p-groups and \(A=S(T)\) then \(R_ A(G)=G(T^*)\). The paper is very readable and contains a useful summary of the concepts of valuated groups, trees, simply presented etc. In Satz 6.7 read \(S(X_ i)\)-free instead of A-free.

Corollary 4.3: Let A be a finite, indecomposable, valuated p-group and G a p-valuated group which possesses a morphism \(\phi\) on G to a group \(\oplus_{J}A\) with \(Ker(\phi)=R_ A(G)\). Then every monomorphism \(\alpha\) : \(\oplus_{I}A\to G\) with \(Im(\alpha)\cap R_ A(G)=0\) splits.

The morphism \(\phi\) always exists for simply presented p-groups A and G. Theorem 6.8: The following statements are equivalent for a simply presented, finite, valuated p-group A.

(a) \(A=\oplus^{n}_{i=1}\oplus_{m_ i}S(X_ i)\) where \(m_ i\) is finite, the \(X_ i\) are finite, irretractable, incomparable, valuated p- trees and S(X) denotes the simply presented valuated p-group with p-basis X.

(b) If P is a direct summand of \(\oplus_{I}A\) then every monomorphism \(\alpha\) on P into a simply presented valuated p-group G splits if and only if \(\alpha (P)\cap R_ A(G)=0.\)

An important tool in the proofs is a category equivalence between the category of direct sums \(\oplus_{I}A\) of a finite valuated p-group A and the category of free Morph(A,A)-modules. Furthermore, a subgroup \(G(T^*)\) of the valuated group G, defined for any valuated p-tree T, is discussed at some length as a possible replacement of the classical subgroup \(p^ nG\). However, when both A and G are simply presented valuated p-groups and \(A=S(T)\) then \(R_ A(G)=G(T^*)\). The paper is very readable and contains a useful summary of the concepts of valuated groups, trees, simply presented etc. In Satz 6.7 read \(S(X_ i)\)-free instead of A-free.

Reviewer: A.Mader

### MSC:

20K10 | Torsion groups, primary groups and generalized primary groups |

20K25 | Direct sums, direct products, etc. for abelian groups |

20K40 | Homological and categorical methods for abelian groups |

### Keywords:

direct summand; finite valuated p-group; p-local valuated group; simply presented, finite, valuated p-group; valuated p-trees; category equivalence
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DOI

### References:

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