Exercises

1. Let $G$ be a group and $S$ a complex of $G$. Let $E$ be the set of all finite products of elements and of inverses of elements of $S$. Prove: $E \leq G$.

2. Show that the relation of being isomorphic, $\cong$, on the class of all groups is an equivalence relation. Describe the equivalence classes of $\cong$.

3. Let $f : G_1 \rightarrow G_2$ be an isomorphism of the group $G_1$ onto the group $G_2$. Let $e_i$ be the identity of $G_i$ ($i = 1, 2$). Prove

   (a) $f(e_1)= e_2$,

   (b) $f(a^{-1}) = f(a)^{-1}$ for all $a\in G_1$.

   HINT: For (a), apply $f$ to both sides of $e_1 e_1 = e_1$. For (b) use (a) together with the definition of inverse.)

4. Prove that any group of prime order is cyclic.

   (Note: If $G = \langle a\rangle$ is a finite cyclic group, $\vert G\vert = o(a)$ - see Property 10 of the elementary properties of groups from Chapter 2.)

5. (a) Show that the group $\mathbb{Z}_n$ of problem 4 for Section 4.3 is cyclic of order $n$.

   (b) Show that $\mathbb{Z}_n \cong G$ where $G$ is the group of Example 5.1.3. (You must show your map is well-defined!)

6. Let $G$ be a group of order $pq$, where $p$ and $q$ are primes such that $p < q$. Prove that $G$ does not contain two distinct subgroups of order $q$.

   HINT: Use a proof by contradiction using Theorem 4.3.6 and problem 4 above.