# Questions tagged [group-homomorphism]

For questions about a function from one group to another that respects the structures of the groups. In symbols, $\varphi$ is a group homomorphisms if for group elements $a$ and $b$, $\varphi(ab)=\varphi(a)\varphi(b)$. Consider also using the broader tags (group-theory) or (abstract-algebra).

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### Question regarding surjectivity of induced homormophism in an old version of Hatcher's proof of Prop. 4.13

So I am currently trying to understand the given proof of Hatcher's proof of proposition 4.13. It's this particular part (in the middle of the screenshot) I don't understand: The extended $f$ ...
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### Determine the number of homomorphisms from $S_{3} \rightarrow \Bbb Z_{2} \times \Bbb Z_{4}$.

Determine the number of homomorphism from $S_{3} \rightarrow \Bbb Z_{2} \times \Bbb Z_{4}$. My attempt: A homomorphism from $S_{3} \rightarrow \Bbb Z_{2} \times \Bbb Z_{4}$ is a homomorphism into ...
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### Suppose there is a group homomorphism φ : G → G0 . If |G| = 1013 and |G0 | = 55, what can you say about the image of φ?

i get that cardinality of image(phi) should divide 55, by lagrange thm. so it can be 1, 5,11 or 55. how to proceed from here
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### If $m \mid n$, show that there is a one-to-one homomorphism $\mathbb{Z}_m \to \mathbb{Z}_n$.

If $m \mid n$, show that there is a one-to-one homomorphism $\mathbb{Z}_m \to \mathbb{Z}_n$. Give an explicit homomorphism $\varphi : \mathbb{Z}_6 \to \mathbb{Z}_{12}$ that is injective, i.e., one-to-...
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### Dual to group algebra of a finite group as a Hopf Algebra

I have a group algebra of a finite group $G$ over $\mathbb{C}$ and $Fun(\mathbb{C}[G])$ represents the linear functions on $\mathbb{C}[G]$. $\mathbb{C}[G] \otimes \mathbb{C}[G]$ denotes the tensor ...