Prove a group of order $48$ must have a normal subgroup of order $8$ or $16$. Solution: The number of Sylow $2$-subgroups is $1$ or $3$. In the first case, there is a normal subgroup of order $16$ so we are done. In the second case, let $G$ act by conjugation on the Sylow $2$-subgroups. […]

I have a question that how to find all homomorphisms from dihedral group $D_{2n}=\langle s,r : s^2=r^n=1 , srs=r^{-1} \rangle$ to the multiplicative group $\mathbb C^\times$? alternating group $A_4$ to the multiplicative group $\mathbb C^\times$? I came up with a single approach to deal with the problems. Note that $D_{2n}$ which is generated by $r, […]

There is finitely generated group $G$ which I don’t know. For every finite group $H$ I can think of, I know the number of homomorphisms $G \to H$ up to conjugation. (By this I mean that two homomorphisms $\phi_1$ and $\phi_2$ are being considered equivalent if there is a $h \in H$ such that $\phi_1(g)h […]

This question already has an answer here: The alternating group is a normal subgroup of the symmetric group 1 answer

Let $A$ be a finitely generated abelian group and $f:A\rightarrow A$ a surjective homomorphism. How do I prove that $f$ is an isomorphism? And if $f$ were injective instead of surjective would the statement still hold?

The problem is to find every group homomorphism $$\varphi:(\mathbb{Z}\times \mathbb{Z}, +,-,0)\longrightarrow(\mathbb{Q}, +,-,0)$$ and the other way around $$\varrho: (\mathbb{Q}, +,-,0) \longrightarrow(\mathbb{Z}\times \mathbb{Z}, +,-,0) $$ I started with the latter one, but I found only one: $\forall x\in\mathbb{Q}: \varrho(x)=(0,0)$ I tried this one: 2. $\varrho(\frac{a}{b})=(a,b), \frac{a}{b}\in\mathbb{Q}$ but it doesn’t work, because $$\varrho(\frac{a}{b})+\varrho(\frac{c}{d})=(a+c,b+d)=\varrho(\frac{a+c}{b+d})\not=\varrho(\frac{a}{b}+\frac{c}{d})$$ I couldn’t think of […]

I need to prove that rings $End(\mathbb{Z}^{n})$ and $M_{n}(\mathbb{Z})$ are isomorphic. To start with, I let $A = \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots& & \vdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} \end{pmatrix} \in M_{n}(\mathbb{Z})$ (so that all the […]

Problem: If $N$ is a normal subgroup of order $p$ where $p$ is the smallest prime dividing the order of a finite group $G$, then $N$ is in the center of $G$. Solution: Since $N$ is normal, we can choose for $G$ to act on $N$ by conjugation. This implies that there is a homomorphism […]

Does there exist any surjective group homomrophism from $(\mathbb R^* , .)$ ( the multiplicative group of non-zero real numbers ) onto $(\mathbb Q^* , .)$ ( the multiplicative group of non-zero rational numbers ) ?

Outside of the technical definitions, what exactly is a homormorphism or an isomorphism “saying”? For instance, let’s we have a group or ring homomorphism $f$, from $A$ to $B$. Does a homomorphism mean that $f$ can send some $a_i$ in $A$ to $b_j$ in $B$, but has no way to “get it back”? Similarly, if […]

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