Question

If $\mathrm{G}$is an abelian group of order $\mathrm{n}$and $\overline{)\mathrm{k}}\mathrm{n}$, prove that there exist a group role="math" localid="1657362262181" $\mathrm{H}$of order role="math" localid="1657362276619" $\mathrm{k}$ and a surjective homomorphism role="math" localid="1657362248490" $\mathrm{G}\to \mathrm{H}$ .

Short answer

It is proved that, there exist a group $\mathrm{H}$of order$\mathrm{k}$ and a surjective homomorphism $\mathrm{G}\to \mathrm{H}$ .

Step-by-step solution

Fundamental Theorem of finite abelian group and Theorem 9.9

Fundamental Theorem of Finite Abelian Group

Every finite abelian group $\mathrm{G}$ is the direct sum of cyclic groups, each of prime power orders.

Theorem 9.9

If $\mathrm{n}={{\mathrm{p}}_1}^{{\mathrm{n}}_1}{{\mathrm{p}}_2}^{{\mathrm{n}}_2}....{{\mathrm{p}}_{\mathrm{t}}}^{{\mathrm{n}}_{\mathrm{t}}}$ with ${\mathrm{p}}_1...{\mathrm{p}}_{\mathrm{t}}$ distinct primes then, ${\mathrm{\mathbb{Z}}}_{\mathrm{n}}\cong {\mathrm{\mathbb{Z}}}_{{{\mathrm{p}}_1}^{{\mathrm{n}}_1}}\oplus ...\oplus {\mathrm{\mathbb{Z}}}_{{{\mathrm{p}}_1}^{{\mathrm{n}}_1}}$.

Given that $\mathrm{G}$ is an abelian group of order $\mathrm{n}$ and $\overline{)\mathrm{k}}\mathrm{n}$.

A group  H of order k 

Let $\mathrm{G}={\oplus }_{\mathrm{i}=1}^{\mathrm{r}}{\mathrm{\mathbb{Z}}}_{{{\mathrm{p}}_{\mathrm{i}}}^{{\mathrm{a}}_{\mathrm{i}}}}$ be an finite abelian group.

Each cyclic group in $\mathrm{G}$ generated by ${\mathrm{a}}_{\mathrm{i}}$ and the order of $\mathrm{n}$ is $\mathrm{n}=\prod _{\mathrm{i}=1}^{\mathrm{r}}{{\mathrm{p}}_{\mathrm{i}}}^{{\mathrm{a}}_{\mathrm{i}}}$.

Since $\overline{)\mathrm{k}}\mathrm{n},\mathrm{k}=\prod _{\mathrm{i}=1}^{\mathrm{r}}{{\mathrm{p}}_{\mathrm{i}}}^{{\mathrm{\beta }}_{\mathrm{i}}}$ with ${\mathrm{\beta }}_{\mathrm{i}}\le {\mathrm{\alpha }}_{\mathrm{i}}$ for each $\mathrm{i}$.

Thus, there exists a subgroup has order $\frac{\mathrm{n}}{\mathrm{k}}=\prod _{\mathrm{i}=1}^{\mathrm{r}}{{\mathrm{p}}_{\mathrm{i}}}^{{\mathrm{\alpha }}_{\mathrm{i}}-{\mathrm{\beta }}_{\mathrm{i}}}$and a surjective homomorphism $\mathrm{\varphi }:\mathrm{G}\to \mathrm{G}/\mathrm{K}$such that $\left|\mathrm{G}/\mathrm{K}\right|=\mathrm{k}$.

Therefore, if $\mathrm{G}$is an abelian group of order $\mathrm{n}$and $\overline{)\mathrm{k}}\mathrm{n}$ then there exist a group $\mathrm{H}$ of order $\mathrm{k}$ and a surjective homomorphism role="math" localid="1657364063191" $\mathrm{G}\to \mathrm{H}$.