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 $𝒫=\{p\in \mathbb{N}|p\text{ is prime}\}$ and ${𝒫}_0$ be any subset of $𝒫$. Then

$$\underset{p\in P_0}{\oplus }{\mathbb{Z}}_p$$

is both Hopfian and co-Hopfian.

Proof. It is easy to see that ${\{{\mathbb{Z}}_p\}}_{p\in 𝒫}$ is full, so ${\{{\mathbb{Z}}_p\}}_{p\in {𝒫}_0}$ 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