For any integers $m, n \in A$, we have
\begin{align*}
\phi(m+n)&=2(m+n)\\
&=2m+2n\\
&=\phi(m)+\phi(n).
\end{align*}
Thus, the map $\phi$ is a group homomorphism.
(b) Prove that $\phi$ is injective.
Suppose that we have
\[\phi(m)=\phi(n)\]
for some integers $m, n\in A$.
This yields that we have $2m=2n$, and hence $m=n$.
So $\phi$ is injective.
Suppose that we have
\[\phi(m)=0.\]
Then we have $2m=0$, and hence $m=0$.
It follows that the group homomorphism $\phi$ is injective.
(c) Prove that there does not exist a group homomorphism $\psi:B \to A$ such that $\psi \circ \phi=\id_A$.
Seeking a contradiction, assume that there exists a group homomorphism $\psi:B \to A$ such that $\psi \circ \phi =\id_A$.
Then we compute
\begin{align*}
&1=\id_A(1)=\psi \circ \phi(1)\\
&=\psi(2)=\psi(1+1)\\
&=\psi(1)+\psi(1) && \text{since $\psi$ is a group homomorphism}\\
&=2\psi(1).
\end{align*}
It yields that
\[\psi(1)=\frac{1}{2}.\]
However note that $\psi(1)$ is an element in $A$, thus $\psi(1)$ is an integer.
Hence we got a contradiction, and we conclude that there is no such $\psi$.
A Homomorphism from the Additive Group of Integers to Itself
Let $\Z$ be the additive group of integers. Let $f: \Z \to \Z$ be a group homomorphism.
Then show that there exists an integer $a$ such that
\[f(n)=an\]
for any integer $n$.
Hint.
Let us first recall the definition of a group homomorphism.
A group homomorphism from a […]
A Group Homomorphism is Injective if and only if the Kernel is Trivial
Let $G$ and $H$ be groups and let $f:G \to K$ be a group homomorphism. Prove that the homomorphism $f$ is injective if and only if the kernel is trivial, that is, $\ker(f)=\{e\}$, where $e$ is the identity element of $G$.
Definitions/Hint.
We recall several […]
A Group Homomorphism is Injective if and only if Monic
Let $f:G\to G'$ be a group homomorphism. We say that $f$ is monic whenever we have $fg_1=fg_2$, where $g_1:K\to G$ and $g_2:K \to G$ are group homomorphisms for some group $K$, we have $g_1=g_2$.
Then prove that a group homomorphism $f: G \to G'$ is injective if and only if it is […]
Inverse Map of a Bijective Homomorphism is a Group Homomorphism
Let $G$ and $H$ be groups and let $\phi: G \to H$ be a group homomorphism.
Suppose that $f:G\to H$ is bijective.
Then there exists a map $\psi:H\to G$ such that
\[\psi \circ \phi=\id_G \text{ and } \phi \circ \psi=\id_H.\]
Then prove that $\psi:H \to G$ is also a group […]
The Quotient by the Kernel Induces an Injective Homomorphism
Let $G$ and $G'$ be a group and let $\phi:G \to G'$ be a group homomorphism.
Show that $\phi$ induces an injective homomorphism from $G/\ker{\phi} \to G'$.
Outline.
Define $\tilde{\phi}([g])=\phi(g)$ and show that this is well-defined.
Show […]
Abelian Normal subgroup, Quotient Group, and Automorphism Group
Let $G$ be a finite group and let $N$ be a normal abelian subgroup of $G$.
Let $\Aut(N)$ be the group of automorphisms of $G$.
Suppose that the orders of groups $G/N$ and $\Aut(N)$ are relatively prime.
Then prove that $N$ is contained in the center of […]
Dihedral Group and Rotation of the Plane
Let $n$ be a positive integer. Let $D_{2n}$ be the dihedral group of order $2n$. Using the generators and the relations, the dihedral group $D_{2n}$ is given by
\[D_{2n}=\langle r,s \mid r^n=s^2=1, sr=r^{-1}s\rangle.\]
Put $\theta=2 \pi/n$.
(a) Prove that the matrix […]