Group Theory Through Problems

A group is an algebraic structure that abstracts the notion of permutations of a set, or more generally, symmetries of an object. Formally a group is a pair $(G, \cdot)$ consisting of a set $G$ and a binary operation $\cdot\colon G \times G \to G$ satisfying the properties or group axioms given below.

1. The operation $\cdot$ is associative -- For all elements $x, y, z \in G$, $x \cdot (y \cdot z) = (x \cdot y) \cdot z$.
2. $G$ has an identity element -- There exists an element $e \in G$, such that for every element $x \in G$, $e \cdot x = x \cdot e = x$.
3. Every element of $G$ has an inverse in $G$ -- For every $x \in G$, there exists a $y \in G$ such that $x \cdot y = y \cdot x = e$.

The second axiom mandates that an empty set not be a group. By definition of $\cdot$ as a binary operation on $G$, it is true that $G$ is closed under $\cdot$ -- i.e. for any two elements $x, y \in G$, the element $x \cdot y$ is also in $G$. Where there is no potential ambiguity, we drop the operator $\cdot$, and simply write $xy$ instead of $x \cdot y$.

A group $G$ is called Abelian if, in addition to the above four axioms, it satisfies the axiom of commutativity. That is, for every pair of elements $x, y \in G$, $xy = yx$.

Exercise 1
Verify that each of the following is a group.

1. $(\mathbb Z, +)$, the set of integers under the operation of addition.
2. $(\mathbb Q, +)$, the set of rational numbers under addition.
3. $(\mathbb Q^*, \times)$, the set of non-zero rational numbers under multiplication.
4. $(\mathbb R, +)$, the set of real numbers under addition
5. $(\mathbb R^*, \times)$, the set of non-zero real numbers under multiplication.
6. $(\mathbb Q_{>0}, \times)$, the set of positive rational numbers under multiplication.
7. $(\mathbb R_{> 0}, \times)$, the set of positive real numbers under multiplication.
8. $(I, \oplus)$, with $I = [0,1)$, and $\oplus$ defined as $$\forall x, y \in I, x \oplus y = (x + y) - \lfloor x + y \rfloor$$ where $\lfloor \cdot \rfloor$ denotes the floor function (greatest integer function).
9. $\operatorname{GL}_n(\mathbb R)$, the set of all real $n \times n$ non-singular matrices, together with the operation of matrix multiplication.
10. $(A^A, \circ)$, the set of all bijective mappings (permutations) from a non-empty set $A$ to itself, under $\circ$, the composition of mappings.

Exercise 2
Which of the groups in Exercise 1 are Abelian?

Exercise 3
Show that the following are not groups.

1. $(\mathbb N_0, +)$, the set of natural numbers together with zero, under addition.
2. $(\mathbb Z, -)$, the set of integers under subtraction.
3. $(\mathbb R, \times)$, the set of all real numbers under multiplication.
4. $(\mathbb Z^*, \times)$, the set of non-zero integers under multiplication.

The group axioms state that every group has an identity element, and every element in a group has an inverse element in the group. But we may notice in all the examples of groups seen so far, that every group has a unique identity element, and every element in the group has a unique inverse. This is, in fact, true for all groups.

Exercise 4
Prove that every group has a unique element $e$ satisfying
$$\forall x \in G, ex = xe = x,$$
and that for each element $x \in G$, there is a unique element $y \in G$ satisfying
$$xy = yx = e.$$
Hint: Assume there is another element $e' \in G$ satisfying the first property, and show that $e = e'$. Similarly, assume that there is another element $z \in G$ satisfying the second property, and show that $y = z$.

Now, we may speak of the identity element of a group, and the inverse of an element of the group. As every element $x$ of a group $G$ has a unique inverse, we may denote it by $x^{-1}$.

Exercise 5
Prove that for any two elements $x$ and $y$ in a group $G$, $(xy)^{-1} = y^{-1}x^{-1}$.

Exercise 6
Prove that a group $G$ is Abelian if $x^{-1} = x$ for all elements $x \in G$.

Thanks to associativity, we can write the product $x^n = \underbrace{x \cdot x \cdots x}_n$ without ambiguity, for any natural number $n$ (why? You are asked to prove this in the exercise below). Generalizing this, we can define $x^0 = e$, the identity element, and $x^{-n} = (x^{-1})^n$. Thus, $x^n$ is well-defined for every integer $n$.

Exercise 7
Prove, by mathematical induction, that the product $\underbrace{x \cdot x \cdots x}_n$ is well-defined. That is, if we define $x^n = \underbrace{x(x(\cdots(x)\cdots))}_n$, for any natural number $n$, show that $x^m \cdot x^n = x^{m + n}$. Also prove that for any integers $m$ and $n$, $x^m \cdot x^n = x^{m + n}$, and that $x^{-n} = (x^n)^{-1}$.

Clearly, it is not true in general that for distinct elements $x$ and $y$ of a group $G$, $(xy)^n = x^n y^n$. But in an Abelian group, this is always true. Indeed, we can even prove a partial converse.

Exercise 8
Prove that a group $G$ is Abelian if and only if for all elements $x, y \in G$, $(xy)^2 = x^2 y^2$.