# 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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### Let $\phi : G\to G'$ be a homomorphism and let $S'\subseteq G'$. Is $\phi^{-1}(\langle S'\rangle ) = \langle \phi^{-1}(S')\rangle$?

I've shown that $\phi^{-1}(\langle S'\rangle ) \supseteq \langle \phi^{-1}(S')\rangle$ and was wondering whether the other inclusion holds, although I've not been able to prove it nor find a ...
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### A link between multiplicative function on endomorphisms and their determinants

Take $E$ a $K$-vector space with $K$ a field with characteristic $0$. Then take $$F:\operatorname{End}_K(E) \to K$$ a map such that $$F(1)=1$$ and $$F(\phi\circ\psi) = F(\phi)F(\psi)$$ Then there ...
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### Prove that $G\cong H\times K$ if and only if there are homomorphisms...

I have some questions for the following following exercise which came from Hungerford's undergraduate Abstract algebra An introduction 3rd edition text in chapter 9, section 1. Let $G$ be an additive ...
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### Why does injectivity imply to $|G/(H \cap K)|\leq |G/H|\cdot|G/K|?$

I'm looking through some proof about the inequality in the title, the one defines: $$\phi: G/(H \cap K)\rightarrow G/H\times G/K$$ $$\phi(g(H \cap K))=(gH,gK)$$ Note that $\phi$ is injective, I'd like ...
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### Draw a covering of $S^1 \vee S^1$ whose fundamental group is isomorphic to $\ker \Phi : F_2 \to \mathbb Z$ with $a\mapsto 2, b\mapsto 3$

$\newcommand{\Z}{\mathbb Z}$ Let $K\leq F_2 = \langle a,b\rangle$ be the kernel of the map $\Phi: F_2 \to \Z$ sending $a$ to $2$ and $b$ to $3$. Draw a cover of $S^1 \vee S^1$ whose fundamental group ...
1 vote
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### Prove that ${\rm Inn}(S_n)$ isomorphic to $S_n.$

Show that ${\rm Inn}(S_n)$ isomorphic to $S_n$ for $n\ge3$. To do this, if I define some isomorphic function say $\phi$, where $\phi: S_n \to{\rm Inn}(S_n)$, then show that $\phi$ is bijective (by ...
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### Is there an element $f$ of $\operatorname{End}(V)$ which is not an element of $\operatorname{End}_K(V)$?
I am reading "An Introduction to Algebraic Systems (in Japanese) by Kazuo Matsuzaka. Let $V$ be a vector space over a field $K$. Let $\operatorname{End}_K(V)$ be the set of all linear mappings ...