Morphic groups.
A group \(G\) is called morphic if every endomorphism \(\alpha\colon G\to G\) for which \(G\alpha\) is normal in \(G\) satisfies \(G/G\alpha\cong\ker(\alpha)\). Main results of the paper under review are as follows: (The numberings are as in the paper.) Theorem 10. Let \(G\) be a uniserial group (i.e., the lattice of normal subgroups is a finite chain), and let \(G=G_0\supset G_1\supset G_2\supset\cdots\supset G_n=1\) denote the lattice of normal subgroups. The following are equivalent: (1) \(G\) is morphic. (2) If \(G/G_k\cong G_{n-k}\), \(k=1,2,\dots,n\), then \(G/G_{n-k}\cong G_k\). Theorem 14. A finitely generated Abelian group is morphic if and only if it is finite and each \(p\)-primary component has the form \((C_{p^k})^n\) for some \(n\geq 0\) and \(k\geq 0\). A group \(G\) is called strongly morphic if it is morphic and every normal subgroup of \(G\) is isomorphic to an image of \(G\). Theorem 18. A finite Abelian group \(G\) is strongly morphic if and only if it is morphic. Theorem 19. Let \(G\) be a strongly morphic group. If \(K\vartriangleleft G\) and \(K_1\vartriangleleft G\), then: \[ G/K\cong G/K_1\text{ if and only if }K\cong K_1. \] Theorem 20. Let \(G\) be a uniserial group, and let \(G=G_0\supset G_1\supset G_2\supset\cdots\supset G_n=1\) denote the normal subgroup lattice. The following are equivalent: (1) \(G\) is strongly morphic. (2) Every normal subgroup of \(G\) is isomorphic to an image of \(G\). (3) \(G/G_k\cong G_{n-k}\) for each \(k=1,2,\dots,n\). (4) Every image of \(G\) is isomorphic to a normal subgroup of \(G\). Theorem 27. A finite nilpotent group is morphic if and only if each Sylow subgroup is morphic. Theorem 34. Let \(P=S_1\times S_2\times\cdots\times S_n\times G\) where each \(S_i\) is a nonabelian simple group, and \(G\) is a morphic group with the descending chain condition on subgroups. If no \(S_i\) is a subquotient of \(G\), then \(P\) is morphic. (A group \(S\) is called a subquotient of a group \(G\) if \(S\) is isomorphic to an image or a subgroup of either a normal subgroup or a factor of \(G\).) Theorem 37. Let \(G\) be a finite, morphic group with \(|G|=p^n\) for some prime number \(p\). Then: (1) All subgroups and images of \(G\) of order \(p^{n-1}\) are isomorphic. (2) \(G/G'\) is morphic. The proofs are elementary.
scientific article; zbMATH DE number 5697065