Exercises

1. In the proof of Theorem 9.2.4, we showed $n\vert t$. Show $m\vert t$ (where these letters have the meaning given there).
2. Let $C_m$ be a cyclic group of order $m$ and $C_n$ be a cyclic group of order $n$ where $gcd (m,n) = 1$. Prove that the generators of $C_{mn} = C_m \times C_n$ are precisely all elements of the form $(a,b)$, where $a$ is a generator of $C_m$ and $b$ is a generator of $C_n$. (HINT: Theorem 1.2.11.)
3. Let $G$ be a finite abelian group of order $n=\prod_{i=1}^k p_i^{\alpha_i}$, where the $p_i$ are distinct primes. Prove that $G = G_1 \times G_2 \times ... \times G_k$, where $G_i$ is the subgroup of $G$ consisting of all elements whose order divides . HINT: Mimic the proof of Theorem 9.2.1.