Exercises

1. If $G_1$ and $G_2$ are finite cyclic groups such that $\vert G_1\vert = \vert G_2\vert = n$ and $f : G_1 \rightarrow G_2$ is defined as in the text. Show $f$ is 1-1, onto, and operation preserving.

2. Let $G = \langle a\rangle$ with $o(a) = n$. Prove that $G$ has a unique subgroup of order $d$ where $d\vert n$.

   HINT: Let $t = n/d$ then $H = \langle a^t\rangle$ is a subgroup of order $d$. Suppose $H_1$ is another subgroup of order $d$. Show $H_1 = H$.

3. Let $G = \langle a\rangle$ with $\vert G\vert = n$. Let $s_1,s_2 > 0$ be integers such that $s_i\vert n$ ($i = 1, 2$). Suppose $H_1=\langle a^{s_1}\rangle$ and $H_2=\langle a^{s_2}\rangle$. Prove that $H_1\cap H_2 = \langle a^{lcm(s_1,s_2)}\rangle$ and $H_1 H_2 = \langle a^{gcd(s_1,s_2)}\rangle$.

   HINT: For the latter, use the fact that the gcd is a linear combination of $s_1$ and $s_2$, i.e., Theorem 1.2.3.)

4. Let $G$ be a group and $a \in G$ such that $o(a) = mn$ where $gcd (m,n) = 1$. Show that one can write $a = bc$ where $o(b) = m$, $o(c) = n$, and $bc = cb$. Moreover, prove the uniqueness of such a representation.

   HINT: Write $mx + ny = 1$, let $b = a^{ny}$, $c = a^{mx}$ and use Theorem 5.2.2. For uniqueness, show using $mx + ny = 1$, that if $b$ and $c$ satisfy the stated conditions in the problem, then they must be given as stated in this hint.

5. Prove that a group of order $p^m$, where $p$ is a prime and $m\in \mathbb{N}$, must contain a subgroup of order $p$.

   HINT: Use Theorem 5.2.1.

6. If in a group $G$ of order $n$, for each positive $d\vert n$, the equation $x^d = 1$ has less than $d + \phi(d)$ solutions, then show $G$ is cyclic.

   HINT: Use the Corollary 5.2.3 and Theorem 5.2.4.

7. (W. Wardlaw) Prove that a finite group $G$ is cyclic if and only if $G$ has no more than $k$ $k-th$ roots of $1$ for every $k\in \mathbb{N}$, where $1$ is the identity of $G$. .

   HINT: A $k-th$ root of $1$ is a solution to $x^k = 1$. Use Theorem 5.2.1 for the only if part and the Corollary 5.2.5 for the if part.