# characterization of full families of groups

Let $\mathcal{G}=\{G_{k}\}_{k\in I}$ be a family of groups. Then $\mathcal{G}$ is full if and only if for any $i,j\in I$ such that $i\neq j$ we have that any homomorphism $f:G_{i}\to G_{j}$ is trivial.

Proof. ,,$\Rightarrow$” Assume that $f:G_{i}\to G_{j}$ is a nontrivial group homomorphism. Then define

 $h:\bigoplus_{k\in I}G_{k}\to\bigoplus_{k\in I}G_{k}$

as follows: if $t\in I$ is such that $t\neq i$ and $g\in\bigoplus_{k\in I}G_{k}$ is such that $g\in G_{t}$, then $h(g)=g$. If $g\in\bigoplus_{k\in I}G_{k}$ is such that $g\in G_{i}$, then $h(g)(j)=f(g(i))$ and $h(g)(k)=0$ for $k\neq j$. This values uniquely define $h$ and one can easily check that $h$ is not decomposable. $\square$

,,$\Leftarrow$” Assume that for any $i,j\in I$ such that $i\neq j$ we have that any homomorphism $f:G_{i}\to G_{j}$ is trivial. Let

 $h:\bigoplus_{k\in I}G_{k}\to\bigoplus_{k\in I}G_{k}$

be any homomorphism. Moreover, let $i\in I$ and $g\in\bigoplus_{k\in I}G_{k}$ be such that $g\in G_{i}$. We wish to show that $h(g)\in G_{i}$.

So assume that $h(g)\not\in G_{i}$. Then there exists $j\neq i$ such that $0\neq h(g)(j)\in G_{j}$. Let

 $\pi:\bigoplus_{k\in I}G_{k}\to G_{j}$

be the projection and let

 $u:G_{i}\to\bigoplus_{k\in I}G_{k}$

be the natural inclusion homomorphism. Then $\pi\circ u:G_{i}\to G_{j}$ is a nontrivial group homomorphism. Contradiction. $\square$

Corollary. Assume that $\{G_{k}\}_{k\in I}$ is a family of nontrivial groups such that $G_{i}$ is periodic for each $i\in I$. Moreover assume that for any $i,j\in I$ such that $i\neq j$ and any $g\in G_{i}$, $h\in G_{j}$ orders $|g|$ and $|h|$ are realitvely prime (which implies that $I$ is countable). Then $\{G_{k}\}_{k\in I}$ is full.

Proof. Assume that $i\neq j$ and $f:G_{i}\to G_{j}$ is a group homomorphism. Then $|f(g)|$ divides $|g|$ for any $g\in G_{i}$. But $f(g)\in G_{j}$, so $|g|$ and $|f(g)|$ are relatively prime. Thus $|f(g)|=1$, so $f(g)=0$. Therefore $f$ is trivial, which (due to proposition) completes the proof. $\square$

Title characterization of full families of groups CharacterizationOfFullFamiliesOfGroups 2013-03-22 18:36:08 2013-03-22 18:36:08 joking (16130) joking (16130) 9 joking (16130) Derivation msc 20A99