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**Quasi-inverse endomorphisms.**
*(English)*
Zbl 1293.20019

The subject of this paper comes from Hopf Galois theory, albeit in an indirect way. Already in the paper of B. Pareigis and the reviewer [J. Algebra 106, 239-258 (1987; Zbl 0615.12026)], Hopf Galois structures on \(G\)-Galois extensions \(L/K\) of fields were described in terms of regular subgroups of \(\text{Perm}(G)\) normalized by \(G\). Childs reformulated this group-theoretical description in a way which is much easier to use, by looking at injections of \(G\) into the holomorphs of groups \(N\) that have the same order as \(G\). Finally in work of Childs and Childs/Corradino, a way was shown how to use so-called Abelian fixed-point-free (fpf) endomorphisms of \(G\) in order to define such injections. (An endomorphism \(\varphi\) of \(G\) is fpf by definition iff \(\varphi(x)=x\) holds only for \(x=e_G\). If \(G\) is an additively written Abelian group, then \(\varphi\) is fpf iff the endomorphism \(\varphi-id_G\) has trivial kernel.) During these reduction steps, the subject becomes more and more explicit. A particular role is played by fpf endomorphisms which are invertible under the circle operation \(\varphi\circ\psi=\psi-\varphi\psi+\varphi\). (In ring theory, this operation, with \(\circ\) replaced by ordinary multiplication, turns the radical into a group.)

One main result of the paper (Theorem 3.4) shows that this invertibility condition on \(\varphi\) (aptly called “quasi-invertibility” by the author) is in suitable context actually equivalent to \(\varphi\) being Abelian (that is, the image of \(\varphi\) is Abelian as a subgroup of \(G\)). This clears up the picture quite a bit. The author also gives a very clear description of fpf Abelian endomorphisms. One first step is an analog of Fitting’s lemma: given an endomorphism \(\varphi\) of a group \(G\) with suitable chain conditions, \(G\) splits up as a semidirect product of \(K\) and \(H\), where \(K\) is the kernel of a sufficiently high power of \(\varphi\) (the “generalized eigenspace”) and \(\varphi\) restricts to an automorphism of \(H\). In Section 5, a complete description of fpf endomorphisms of finite Abelian \(p\)-groups is given. Section 6 describes fpf endomorphism of semidirect products, and this also leads to a proof of Theorem 3.4 via the analog of Fitting’s lemma. The final section takes up examples considered in previous work of Childs and provides many new examples.

[Reviewer’s remark concerning the proof of Prop. 5.1 concerning fpf endomorphisms of Abelian \(p\)-groups \(G=H_1\times\cdots\times H_n\): This uses a kind of matrix representation, assuming that each \(H_i\) is a product of cyclic groups of order \(p^{e_i}\) and the \(e_i\) are strictly increasing. The endomorphism \(\alpha\) of \(G\) is represented as a matrix \((\alpha_{ij})\), and it is shown that \(\alpha\) is fpf iff each diagonal term \(\alpha_{ii}\) induces an fpf endomorphism of the socle of \(H_i\). A little more conceptually one might say this as follows. The point is that \(\gamma:=\alpha-id\) is monic, that is: an isomorphism. Now since the matrix \(A=(\gamma_{ij})\) is lower triangular modulo \(p\), one sees at once that \(A\) is invertible iff its diagonal entries \(\gamma_{ii}\) are. This in turn is equivalent to \(\gamma_{ii}\) being invertible mod \(p\).]

This nice paper exemplifies that clever observations, together with elementary methods, can also lead to interesting new results without using any big machinery.

One main result of the paper (Theorem 3.4) shows that this invertibility condition on \(\varphi\) (aptly called “quasi-invertibility” by the author) is in suitable context actually equivalent to \(\varphi\) being Abelian (that is, the image of \(\varphi\) is Abelian as a subgroup of \(G\)). This clears up the picture quite a bit. The author also gives a very clear description of fpf Abelian endomorphisms. One first step is an analog of Fitting’s lemma: given an endomorphism \(\varphi\) of a group \(G\) with suitable chain conditions, \(G\) splits up as a semidirect product of \(K\) and \(H\), where \(K\) is the kernel of a sufficiently high power of \(\varphi\) (the “generalized eigenspace”) and \(\varphi\) restricts to an automorphism of \(H\). In Section 5, a complete description of fpf endomorphisms of finite Abelian \(p\)-groups is given. Section 6 describes fpf endomorphism of semidirect products, and this also leads to a proof of Theorem 3.4 via the analog of Fitting’s lemma. The final section takes up examples considered in previous work of Childs and provides many new examples.

[Reviewer’s remark concerning the proof of Prop. 5.1 concerning fpf endomorphisms of Abelian \(p\)-groups \(G=H_1\times\cdots\times H_n\): This uses a kind of matrix representation, assuming that each \(H_i\) is a product of cyclic groups of order \(p^{e_i}\) and the \(e_i\) are strictly increasing. The endomorphism \(\alpha\) of \(G\) is represented as a matrix \((\alpha_{ij})\), and it is shown that \(\alpha\) is fpf iff each diagonal term \(\alpha_{ii}\) induces an fpf endomorphism of the socle of \(H_i\). A little more conceptually one might say this as follows. The point is that \(\gamma:=\alpha-id\) is monic, that is: an isomorphism. Now since the matrix \(A=(\gamma_{ij})\) is lower triangular modulo \(p\), one sees at once that \(A\) is invertible iff its diagonal entries \(\gamma_{ii}\) are. This in turn is equivalent to \(\gamma_{ii}\) being invertible mod \(p\).]

This nice paper exemplifies that clever observations, together with elementary methods, can also lead to interesting new results without using any big machinery.

Reviewer: Cornelius Greither (Neubiberg)

### MSC:

20D45 | Automorphisms of abstract finite groups |

20D25 | Special subgroups (Frattini, Fitting, etc.) |

12F10 | Separable extensions, Galois theory |