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### Let $\phi: \mathbb Z_{15} \to \mathbb Z_{15}$ with $\phi(2)=5$. What is $\phi(1)?$

Hint: you have $2\phi(1)=5 =5+15=20$ in $\mathbb Z/15\mathbb Z$.
• 60.7k

### “Logarithm” with respect to Dirichlet convolution

Just as functions $f:\Bbb Z\to\Bbb C$ with pointwise addition and additive convolution (and upward-closed support) correspond to formal Laurent series $\sum f(n)x^n$ with typical addition and ...
• 2,981
Accepted

• 4,437

### Let $\phi: \mathbb Z_{15} \to \mathbb Z_{15}$ with $\phi(2)=5$. What is $\phi(1)?$

You can find $\phi(x),$ since $1$ is not the only generator of $\Bbb Z_{15}.$ There are $8=\varphi (15)$ of them, where $\varphi$ is Euler's phi function. They are $1,2,4,7,8,11,13$ and $14.$ One ...
• 8,547
1 vote
Accepted

### If $G$ is an additive group, $u,v$ endomorphisms, then if $h(x)= x-u(v(x))$ is onto then $f(x)= x-v(u(x))$ is onto

Given $y$, we want to find $x$ such that $(1 - vu)(x) = y$ ($1$ stands for the identity map). By assumption, there exists $z$ such that $(1 - uv)(z) = u(y)$. Now put $x = y + v(z)$, and check that ...
• 8,504
1 vote
Accepted

• 3,099
1 vote

### For what $n$ is there an injective homomorphism from $\mathbb{Z_n} \to S_7$

For $n=1,\dots,7$, clearly $S_7$ contains $n$-cycles: each of them is isomorphic to $\mathbb Z_n$. For $n=10$, for example the subgroup $\langle (12)(34567)\rangle$ is isomorphic to $\mathbb Z_{10}$. ...
• 3,099
1 vote

### When is there an homomorphism into $S_n$?

As pointed out in the other answer, the $G$-action by left multiplication on the left quotient set $G/H$ yields a homomorphism $G\stackrel{\varphi}{\to} S_{G/H}\cong S_4$. But there's more: since \$H\...
• 3,099

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