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Radikale und koradikale Regeln. (German) Zbl 0562.20017
If \({\mathfrak X}\) is a class of groups, a rule on \({\mathfrak X}\) is a function \({\mathfrak r}\) assigning to each \({\mathfrak X}\)-group G a subgroup \({\mathfrak r}(G)\). The rule \({\mathfrak r}\) is called radical (respectively coradical) if whenever \(S\leq T\) and S,T\(\in {\mathfrak X}\), one has \(S\cap {\mathfrak r}(T)\leq {\mathfrak r}(S)\) (respectively \(S\cap {\mathfrak r}(T)\geq {\mathfrak r}(S))\). The rule \({\mathfrak r}\) is called functorial if \(\alpha\) (\({\mathfrak r}(G))={\mathfrak r}(\alpha (G))\) for every isomorphism \(\alpha\) from an \({\mathfrak X}\)-group G.
The author discusses properties and constructs examples of functorial rules which are radical or coradical. For example, if \({\mathfrak X}\) is an image closed class and \({\mathfrak r}_ 1,{\mathfrak r}_ 2\) are radical functorial rules on \({\mathfrak X}\), define a new rule \({\mathfrak r}_ 2\circ {\mathfrak r}_ 1\) by \({\mathfrak r}_ 2\circ {\mathfrak r}_ 1(G)/{\mathfrak r}_ 1(G)={\mathfrak r}_ 2(G/{\mathfrak r}_ 1(G))\). Then \({\mathfrak r}_ 2\circ {\mathfrak r}_ 1\) is a radical functorial rule on \({\mathfrak X}\). There is a similar result for coradical functorial rules.
Reviewer: D.J.S.Robinson

MSC:
20E07 Subgroup theorems; subgroup growth
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References:
[1] O.H. Kegel - B. A. F. WEHRFRITZ, Locally finite groups , North Holland , Amsterdam ( 1973 ). MR 470081 | Zbl 0259.20001 · Zbl 0259.20001
[2] D.J.S. Robinson , Finiteness conditions and generalized sotubte groups , Part 1, Springer , Berlin ( 1972 ). Zbl 0243.20032 · Zbl 0243.20032
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