# groups question

• Apr 25th 2010, 04:19 PM
wutang
groups question
Can there be a homomophism f, from Z16 onto Z2+Z2?
To figure this out I tried to use the properties of subgroups of homomorphisms. If H is a subgroup Z16, then f(H) is cyclic. But all of the subgroups of Z2+Z2 are also cyclic so I am having a hard time getting a contradiction. Can anyone explain to me were I am going wrong on this?
Thanks
• Apr 25th 2010, 07:19 PM
tonio
Quote:

Originally Posted by wutang
Can there be a homomophism f, from Z16 onto Z2+Z2?
To figure this out I tried to use the properties of subgroups of homomorphisms. If H is a subgroup Z16, then f(H) is cyclic. But all of the subgroups of Z2+Z2 are also cyclic so I am having a hard time getting a contradiction. Can anyone explain to me were I am going wrong on this?
Thanks

Suppose $\phi:\,\mathbb{Z}_{16}\rightarrow \mathbb{Z}_2\times\mathbb{Z}_2$ is an epimorphism $\Longrightarrow \mathbb{Z}_{16}/\ker(\phi) \cong \mathbb{Z}_2\times \mathbb{Z}_2$ , which of course is a contradiction since any homomorphic image of a cyclic

group is cyclic ( in our case, if $\mathbb{Z}_{16}=\,,\,\,then\,\,\,\mathbb{Z}_{16 }/\ker(\phi)=$ )

Tonio