Articles of group homomorphism

Does an injective $\mathbb F_9$ vector space homomorphism $\mathbb F_9^3 \to \mathbb F_9^5$ exist?

Does an injective $\mathbb F_9$ vector space homomorphism $\mathbb F_9^3 \to \mathbb F_9^5$ exist? Is it able to solve that task by some technique? If so, how is it working then? I have posted a similiar question here but with a mapping that is not a homomorphism.

Find all homomorphisms from $D_{2n}$ to $\mathbb C^\times$ (revisit)

I actually was asking the same question in here but haven’t gotten any feedback yet. I now can elaborate a little so that final answer would be closer. I wanted to find all homomorphisms from the dihedral group $D_{2n}=\langle s,r : s^2=r^n=1 , srs=r^{-1} \rangle$ to the multiplicative group $\mathbb C^\times$. Note that $D_{2n}$ which […]

Fastest way to show that $D_6 \to S_5$ is an injective homomorphism

I want to show that there is an injective homomorphism from $D_6 \to S_5$ where $D_6$ denotes the dihidral group of order 12 and $S_5$ the symmetric group. But I’m not sure how I can do this efficiently. I define $f: D_6 \to S_5$ by $f(\sigma) = (12)$ and $f(\rho) = (123)(45)$, with $\sigma$ being […]

What automorphisms exist on the abelian group of positive rationals under multiplication?

Consider the abelian group $(\mathbb{Q}_{>0}, \times)$. What automorphisms exist for this group? I can only think of the trivial one and of $\phi(q) = \frac{1}{q}$. If we relax the problem to injective homomophisms from $(\mathbb{Q}_{>0}, \times)$ to itself, do we get additional results?

Abstract algebra: for a polynomial $p$, prove $\sigma(\tau(p))=(\sigma\tau)(p)$ for all $\sigma, \tau \in S_n$

I’m trying to solve to following problem: Part 1: Let $\{x_1,…x_n\}$ be variables. For any polynomial $p$ in $n$ variables and for $\sigma$ $\in S_n$ define $\sigma (p)(x_1,…,x_n)=p(x_{\sigma(1)},…,x_{\sigma(n)}$ Check that $\sigma(\tau(p))=(\sigma\tau)(p)$ for all $\sigma, \tau \in S_n$ My attempt: $\sigma(\tau(p))=\sigma(p(x_{\tau(1)},…,x_{\tau(n)}))=p(x_{\sigma(\tau(1))},…,x_{\sigma(\tau(n))})=(\sigma\tau)(p)$ Part 2: Fix $n$ and define $\Delta=\prod_{1\leq i < j \leq n} (x_i-x_j)$. Check that […]

How to apply 1st Isomorphism Theorem to show that a finite abelian group has a subgroup of order $n$ for each positive divisor $n$ of its order

I know this question has been posted previously here and here, but they only help with the basic structure of the proof. I feel pretty comfortable up until the part where I start using the 1st Isomorphic Theorem. I just don’t see how this how the preimage and this tie together. I know this is […]

Proof involving homomorphism between $\Bbb Z^n$ and an abelian group G

This question already has an answer here: Question on group homomorphisms involving the standard Z-basis [closed] 1 answer

Group of order $48$ must have a normal subgroup of order $8$ or $16$

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. […]

Find all homomorphisms from $A_4, D_{2n}$ to $\mathbb C^\times$

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, […]

Can I recover a group by its homomorphisms?

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 […]