Exercises

1. If $H \leq G$, a group, and $a,b \in H$, show that the mapping $f : aH\rightarrow bH$ defined by $f(ah) = bh$, for any $h\in H$, is 1-1 and onto.

2. Prove that the map in (4.7) is 1-1 and onto.

3. Verify the other two group axioms for the multiplication of equivalence classes in ${\cal R} (m)$ defined by (4.12) (i.e., associativity and existence of an identity element).

4. Let $\mathbb{Z}_m = \{[0], ..., [m-1]\}$, for $m$ a positive integer, where $[x]$ is the equivalence class with respect to the equivalence relation of congruence modulo $m$. (Later, we shall also denote $\mathbb{Z}_m$ by the notation $\mathbb{Z}/m\mathbb{Z}$ - see Example 6.2.1 below.)

   (a) Show that $\mathbb{Z}_m$ contains all the equivalence classes.

   (b) Show that addition of equivalence classes defined by $[n] + [k] = [n + k]$ is a well defined operation on $\mathbb{Z}_m$.

   (c) Show that multiplication of equivalence classes defined by $[n] \cdot [k] = [n \cdot k]$ is a well defined operation on $\mathbb{Z}_m$.

   (d) Finally show $\mathbb{Z}_m$ is a group with respect to the $+$ operation.

   Use Theorem 1.2.12 above.

5. Find the left and right coset decompositions (partitions) of $S_3$ with respect to all of its subgroups. (See problem 3 from Section 3.1.)

6. Let $G$ be an abelian group of order $6$. Show that there exists an element $a \in G$ such that $G = \{e, a, a^2, a^3, a^4,a^5\}$, i.e., $o(a) = 6$.

   (HINT: Use Corollary 4.3.3 of Lagrange's Theorem first to determine the possible orders of elements in $G$. Next show that if $G$ has more than one element of order $2$, then $G$ must have a subgroup of order $4$. This is a contradiction (why?). Thus $G$ can only have at most one element of order $2$, say $x$. Similarly, show $G$ can have at most one element of order $3$, say $y$. Let $y\in G$ such that $y \notin \{e,x\}$ but $o(y) = 3$. Show this implies $G$ must have an element of order $6$ by considering $xy$.)

7. Suppose $G$ is a finite group with precisely $2$ conjugacy classes. Prove $\vert G\vert = 2$.

   (HINT: Decompose $G$ into conjugacy classes, where one of the classes is the $Cl(e)$. Write an equation for the $\vert G\vert$ from this decomposition - like the class equation (4.10). What is $\vert Cl(e)\vert$? Next use Theorem 4.3.4 to find the order of the other conjugacy class. Finally, use Lagrange's Theorem 4.3.1, in particular equation (4.9), to write this in terms of $\vert G\vert$. Solve your equation for $\vert G\vert$ and use this to prove $\vert G\vert = 2$.)