6 votes
Accepted

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 $\mathbb Z/15\mathbb Z$.
J. W. Tanner's user avatar
  • 60.7k
5 votes

“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 ...
coiso's user avatar
  • 2,981
4 votes
Accepted

Showing that homomorphism is uniquely determined by its image on $X$

As mentioned in the comments, it's not necessary true that every element is a product of elements of $X$, but a product of elements of $X$ and their inverses. Note that for each $x\in X$ we have $\phi(...
Mark's user avatar
  • 40.2k
2 votes

Let $G$ be a finite abelian group of order n. How many distinct group homomorphisms are there from $G$ to $R/Z$?

The question does indeed turn out to be well-defined, but this is not obvious and the "order $n$" part makes it a little confusing anyway as it's much easier to answer for all $n$ at once. ...
hunter's user avatar
  • 30.3k
2 votes

Let $G$ be a finite abelian group of order n. How many distinct group homomorphisms are there from $G$ to $R/Z$?

This gets you into group cohomology. For any subgroup $A\le G$ such that $G/A$ is cyclic, it's the number of central extensions $$0\to A\to G\to G/A\to 0,$$ by the first isomorphism theorem. These ...
calc ll's user avatar
  • 8,547
2 votes

Seifert-van Kampen theorem, classical version

It tells you: $j_1:U\subset X$ and $j_2:V\subset X$ induce maps on $\pi_1$, $\pi_1(U)\to\pi_1(X)$ and $\pi_1(V)\to\pi_1(X)$, and these are genuine homomorphisms, and if you have two groups $A,B$ and a ...
FShrike's user avatar
  • 40.5k
2 votes

Uniqueness in Seifert-van Kampen theorem in Munkres' Topology

This is the obvious consequence of the property that $\Phi \circ j_k = \phi_k$ for $k = 1, 2$. $\pi_1(X,x_0)$ is generated by the set $\Gamma = j_1(\pi_1(U,x_0)) \cup j_2(\pi_1(V,x_0))$. For each $g_1 ...
Kritiker der Elche's user avatar
2 votes

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 ...
calc ll's user avatar
  • 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 ...
user43208's user avatar
  • 8,504
1 vote
Accepted

Restricting a homomorphism $\mathbb{Q}/\mathbb{Z}\rightarrow \overline{\mathbb{F}}_p^\times$

We only need to consider homomorphisms in the category $Ab$ of abelian groups, so moving over there. A useful fact is that the multiplicative group $k^*$ of the algebraic closure $k:=\overline{\Bbb{F}...
Jyrki Lahtonen's user avatar
1 vote
Accepted

Huppert, III.19.2: How to construct a homomorphism from a $p$-group into the center of a maximal subgroup?

So, to try to make such a morphism (somewhat) explicit, we have little choice but to start with the canonical projection $\pi:G\to G/N\cong \Bbb Z_p.$ Then since $N$ is a $p$-group, it has non-...
calc ll's user avatar
  • 8,547
1 vote

How many homomorphisms are there from $D_5$ to $V_4$?

The commutator of $D_5$ is $\langle r\rangle \cong \Bbb Z_5.$ As a result it's in the kernel (there's two ways to see that: $V_4$ has order $4,$ and it's abelian) , and the homomorphisms have every ...
calc ll's user avatar
  • 8,547
1 vote

How many homomorphisms are there from $D_5$ to $V_4$?

Since $2\nmid 5$, all the four elements of order $5$ must be sent to $1$. As for the five elements of order $2$, $\varphi(sr^k)=$ $\varphi(s)\varphi(r^k)=$ $\varphi(s)$. Therefore, for $V_4=\{1,a,b,ab\...
Kan't's user avatar
  • 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 $\mathbb Z_{10}$. ...
Kan't's user avatar
  • 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\...
Kan't's user avatar
  • 3,099

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