More Types of Groups

Prove that

for any positive integers $m$ and $n$ without using the isomorphism theorems.

The number 12 is equivalent to 0 in the group $\mathbb{Z}_{n}$. What are the possible value(s) of $n$?

In the group $\mathbb{Z}_{210}$, find the order of 10, 42, 75, and 140.

Find all the generators of the following groups.

1. $\mathbb{Z}_{10}$

2. $\mathbb{Z}_{101}$

Find all the normal subgroups of the quaternion group $\mathrm{Q}$.(Hint: consider 1.8.)

Write the transposition $\begin{pmatrix}2&6\end{pmatrix}$ into a composition of adjacent transpositions.

Find $\mathrm{sgn}\left(\begin{pmatrix}1&3&2&5&4\end{pmatrix}\right)$.

List all elements of $\mathrm{A}_{3}$ and $\mathrm{A}_{4}$.

List the elements of $\mathcal{U}_{10}$.

Let $a\in\mathcal{U}_{n}$. Prove that $|a|$ divides $\varphi(n)$.

Find the matrix given by the product

Let $G$ be a group. Prove that $\mathrm{Inn}(G)\unlhd\mathrm{Aut}(G)$.

Consider the group $\mathbb{Z}_{10101}$.

1. Find the smallest positive integer $a$ such that $1870a$ is a multiple of 10101.

2. Find the smallest positive integer $b$ such that $3774b$ is a multiple of 10101.

Find the largest integer $n$ such that $\mathrm{A}_{n}$ is an abelian group, proving your claim. Hence find all integers $k$ such that $\mathrm{A}_{k}$ is cyclic.

Suppose $r$ is an odd primitive root modulo $p^{k}$, where $p$ is an odd prime and $k\geq 1$. Prove that $r$ is also a primitive root modulo $2p^{k}$.

Let $G$ be a cyclic group of order $n$ with generator $g$.

1. Suppose the maps $f_{1}:G\to G$ and $f_{2}:G\to G$ are homomorphisms. Prove that $f_{1}=f_{2}$ if and only if $f_{1}(g)=f_{2}(g)$.

2. Let the map $f:G\to G$ be a homomorphism. Explain why $f(g)=g^{m_{f}}$ for some $m_{f}\in\mathbb{Z}_{n}$.

3. Show that the $m_{f}$ obtained in (ii) is unique to $f$.

4. Suppose the functions $f_{1}:G\to G$ and $f_{2}:G\to G$ are homomorphisms. Prove that

   where $\circ$ denotes function composition and $\otimes_{n}$ denotes multiplication modulo $n$.

5. Let the function