Direct decompositions of ideals and normal subgroups
Using some rather unusual techniques (involving a careful analysis of the action of certain sets of operators on subjacent subgroups of skew rings), the author obtains two quite general results (Theorems (2.5) and (3.2)) on direct decompositions of groups. Their statements are too technical to be reproduced here; however, to give an idea of what happens, note that by suitable choices of the set of operators, Maschke's problem and a result of Nakayama [Th. VII 9.3 of ``Finite Groups II'' (1982; Zbl 0477.20001) by \textit{B. Huppert} and \textit{N. Blackburn}] are obtained as particular cases, as well as the classical Krull-Schmidt theorem. Due to the techniques used, these results apply to groups as well as to rings, their particular versions for algebras over fields being of interest from the representation point of view. We state here only two of their corollaries, in their version for groups. Theorem I. Let G be a finite group, H an abelian normal subgroup of G and \(W\trianglelefteq X\leq N_{Aut(G)}(H)\) such that \((| X/W|,| H|)=1\). Then among the maximal W-invariant normal subgroups of G contained in H and having a W-invariant complement in G there is one which is X-invariant and has an X-invariant complement in G. Theorem II. Let G be a finite group and \(W\trianglelefteq X\leq Aut(G)\) such that \((| X/W|,| G/G'|,| Z(G)|)=1\). Then: i) G is the X-direct product of an abelian normal subgroup and a normal subgroup without any W-invariant direct abelian nontrivial factor. ii) If G is X-direct indecomposable, then G is the direct product of isomorphic W-invariant and W-indecomposable subgroups. If, moreover, G is nonabelian, then there exists a direct decomposit\({}_ 0\) which label \(\alpha\). The entire lattice \({\mathcal L}\) may be treated by a slight extension of this theorem, if so desired. Two examples may illuminate the power of this theorem. (1) For any variety \({\mathcal V}\) of bands, \(\underline {\mathcal V}={\mathcal T}\), the trivial variety. Thus all ladders for band varieties are labelled by \({\mathcal T}\), \({\mathcal L}\), \({\mathcal R}\) or \({\mathcal O}\). The (known) lattice of band varieties can now be very easily recovered. (2) For any variety \({\mathcal V}\) of orthogroups (that is, of completely regular semigroups whose idempotents form a subsemigroup), \(\underline {\mathcal V}\) is a group variety. The lattice of all orthogroup varieties is completely determined in terms of the lattice of group varieties, solving a longstanding problem. In the final section, which readers should peruse earlier, the operators and congruences defined above are shown to coincide with those studied by \textit{F. Pastijn} [``The lattice of completely regular semigroup varieties'' (to appear)], by \textit{N. R. Reilly} [J. Aust. Math. Soc., Ser. A 38, 372-393 (1985; Zbl 0572.20040)] and by the reviewer [J. Aust. Math. Soc., Ser. A 42, 227-246 (1987; Zbl 0613.20038)], all independently. (This fact was also discovered by Pastijn, who thereby deduced the modularity of the lattice \({\mathcal L}\) from that of \({\mathcal E}).\) The paper contains a host of other applications, including a detailed analysis of the interactions amongst these operators and others considered by various authors.
scientific article; zbMATH DE number 4040170
scientific article