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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:
 2e+08 Subgroup theorems; subgroup growth
##### Keywords:
radical functorial rules; coradical functorial rules
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##### References:
  O.H. Kegel - B. A. F. WEHRFRITZ, Locally finite groups , North Holland , Amsterdam ( 1973 ). MR 470081 | Zbl 0259.20001 · Zbl 0259.20001  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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