# full families of Hopfian (co-Hopfian) groups

Proposition^{}. Let ${\{{G}_{i}\}}_{i\in I}$ be a full family of groups. Then each ${G}_{i}$ is Hopfian (co-Hopfian) if and only if ${\oplus}_{i\in I}{G}_{i}$ is Hopfian (co-Hopfian).

Proof. ,,$\Rightarrow $” Let

$$f:\underset{i\in I}{\oplus}{G}_{i}\to \underset{i\in I}{\oplus}{G}_{i}$$ |

be a surjective^{} (injective^{}) homomorphism^{}. Since ${\{{G}_{i}\}}_{i\in I}$ is full, then there exists family of homomorphisms ${\{{f}_{i}:{G}_{i}\to {G}_{i}\}}_{i\in I}$ such that

$$f=\underset{i\in I}{\oplus}{f}_{i}.$$ |

Of course since $f$ is surjective (injective), then each ${f}_{i}$ is surjective (injective). Thus each ${f}_{i}$ is an isomorphism^{}, because each ${G}_{i}$ is Hopfian (co-Hopfian). Therefore $f$ is an isomorphism, because

$${f}^{-1}=\underset{i\in I}{\oplus}{f}_{i}^{-1}.\mathit{\hspace{1em}}\mathrm{\square}$$ |

,,$\Leftarrow $” Fix $j\in I$ and assume that ${f}_{j}:{G}_{j}\to {G}_{j}$ is a surjective (injective) homomorphism. For $i\in I$ such that $i\ne j$ define ${f}_{i}:{G}_{i}\to {G}_{i}$ to be any automorphism of ${G}_{i}$. Then

$$\underset{i\in I}{\oplus}{f}_{i}:\underset{i\in I}{\oplus}{G}_{i}\to \underset{i\in I}{\oplus}{G}_{i}$$ |

is a surjective (injective) group homomorphism. Since ${\oplus}_{i\in I}{G}_{i}$ is Hopfian (co-Hopfian) then ${\oplus}_{i\in I}{f}_{i}$ is an isomorphism. Thus each ${f}_{i}$ is an isomorphism. In particular ${f}_{j}$ is an isomorphism, which completes^{} the proof. $\mathrm{\square}$

Example. Let $\mathcal{P}=\{p\in \mathbb{N}|p\text{is prime}\}$ and ${\mathcal{P}}_{0}$ be any subset of $\mathcal{P}$. Then

$$\underset{p\in {P}_{0}}{\oplus}{\mathbb{Z}}_{p}$$ |

is both Hopfian and co-Hopfian.

Proof. It is easy to see that ${\{{\mathbb{Z}}_{p}\}}_{p\in \mathcal{P}}$ is full, so ${\{{\mathbb{Z}}_{p}\}}_{p\in {\mathcal{P}}_{0}}$ is also full. Moreover for any $p\in {P}_{0}$ the group ${\mathbb{Z}}_{p}$ is finite, so both Hopfian and co-Hopfian. Therefore (due to proposition)

$$\underset{p\in {P}_{0}}{\oplus}{\mathbb{Z}}_{p}$$ |

is both Hopfian and co-Hopfian. $\mathrm{\square}$

Title | full families of Hopfian (co-Hopfian) groups |
---|---|

Canonical name | FullFamiliesOfHopfiancoHopfianGroups |

Date of creation | 2013-03-22 18:36:05 |

Last modified on | 2013-03-22 18:36:05 |

Owner | joking (16130) |

Last modified by | joking (16130) |

Numerical id | 7 |

Author | joking (16130) |

Entry type | Theorem |

Classification | msc 20A99 |