Introduction to groups

There are no precise boundaries between different branches of math, no strict and formal rules defining which concepts and facts belong to discrete math and which to algebra, which to real analysis, and which to the topology. But a somewhat vague description of algebra may say that it's an area of math seeking similar patterns in different situations.

For example, one of the earliest algebraic ideas, introduced by the 16th-century French mathematician François Viète, was representing known and unknown numbers by letters. Now it seems natural to us to write $(a + b)^{2} = a^{2} + 2ab + b^{2}$ without bothering ourselves about what numbers hide under $a$ and $b$, but four and a half centuries ago, using letters was a breakthrough. (Fun fact: Viète served as a codebreaker to a French king, and when the King of Spain discovered that his secret communications were read, he was shocked and accused the French of using black magic.)

Looking at the further development of algebra, you notice that it always tries to find some general structures, to reveal similarities. These general patterns may seem abstract, but that makes them more universal. Let's talk about the concept of a group -- abstract, but simple, ubiquitous, and, as it turns out, very useful.

Binary operations

Consider the integers. We can add, subtract and multiply them, we can also find the greatest common divisor of any two of them. In all these cases, we take two integers as our input and produce a new integer as an output. That means we have just seen several examples of binary operations (on the set $\mathbb{Z}$).

Definition. A binary operation on a set $A$ is a mapping (a function) from $A \times A$ to $A$.

The result of a binary operation applied to $(a, b)$ is usually written in the form $a \star b$ , where a star is a special symbol for the operation. In different settings, the symbols can be different, such as plus $+$, dot $\cdot$ or circle $\circ$. Moreover, sometimes a symbol of the operation(usually the symbol $\cdot$ ) is omitted, and instead of $a \cdot b$ we write $ab$ . The arguments $a$ and $b$ are called the operands.

Let's consider one more example. Suppose $X$ is a non-empty set and $S(X)$ is the set of all bijections from $X$ to itself. For instance, if $X = \{1, 2\}$, then $S(X)$ consists of only two bijective mappings: $S(X) = \{f, g\}$ (look at the pictures):

Of course, for larger $X$, the set $S(X)$ will also be much larger. But no matter how tiny or how huge $X$ is, for any $f, g \in S(X)$ we can form the composition $g \circ f$ defined as usual: if $x \in X$, then $$(g \circ f)(x) = g(f(x))$$So the composition is, in fact, a binary operation on $S(X)$.

The definition of a group

Can we notice some common properties for addition on the set of integers $\mathbb{Z}$ and composition on the set of bijections $S(X)$ for some $X$?

First, both operations are associative. It doesn't matter how you place the brackets in the expression $$2 + 3 + 4,$$as both $$(2 + 3) + 4 = 5 + 4$$ and $$2 + (3 + 4) = 2 + 7$$ give $9$, and the similar fact is true for any three integers. The brackets' placement also doesn't change anything with functions:

$$h \circ (g \circ f) =(h \circ g) \circ f$$If you wonder why the last statement is true, try to apply the function from the left-hand side and the function from the right-hand side to the same element $x \in X$ and compare the results:

$$(h \circ (g \circ f))(x) = h ((g \circ f)(x)) = h(g(f(x))\\$$

and

$$((h \circ g) \circ f)(x) = (h \circ g) (f(x)) = h(g(f(x))\\$$

Second, adding zero to any integer doesn't change this integer. An analogous role in $S(X)$ will be played by an identity function $id \in S(X)$ such that

$$id(x) = x \text{ for every } x \in X$$Indeed, for any $f \in S(X)$ and any $x \in X$

$$(id \circ f) (x) = id(f(x)) = f(x)\\$$$$(f \circ id)(x) = f(id(x)) = f(x)$$so $$id \circ f = f \circ id = f.$$Third, for any integer $a$, the sum of $a$ and $-a$ is zero. Likewise, any bijection $f$ from $S(X)$ is invertible -- that means, there exists another function, the composition of which with $f$ (in any order) gives the identity function.

Let's give a formal definition.

Definition. A set $G$ with a binary operation $\star$ on it is called a group, if it satisfies the following three properties:

1. $(a \star b) \star c = a \star (b \star c)$ for any $a, b, c \in G$ (this property is known as the associativity of operation $\star$)
2. there exists an element $e \in G$ (called a neutral element or an identity) such that $a \star e = e \star a = a$ for any $a \in G$
3. for each $a$ there exists an element $a' \in G$ such that $a \star a' = a' \star a = e$ (such $a'$ is called an inverse of $a$)

Note that the neutral element is unique: if both $e$ and$e'$ are identities, then $$e = e \star e' = e.$$Also, the inverse of a given $a$ is unique.

Indeed, suppose that $a'$ and $a''$ are both inverses of $a$. That means that $a \star a'' = e$ and $a' \star a = e$, so

$$a' = a' \star e =a' \star (a \star a'') = (a'\star a) \star a'' = e \star a'' = a''$$

We'll usually denote set $G$ with a binary operation $\star$ , satisfying the definition of a group, as $(G, \star)$. In some situations, mathematicians specify only the set and assume that the operation is obvious from the context.

More examples

Of course, the examples of groups are not limited to $(\mathbb{Z}, +)$ and $(S(X), \circ)$. Consider the set $\mathbb{R}^{*} = \mathbb{R} \setminus \{0 \}$ (that is, all real numbers except zero) and multiplication on it. You can easily see that the operation is associative, that $1$ is a neutral element and that for every element $r \in \mathbb{R}^{*}$ -- in other words, for any real number not equal to zero -- there exist an inverse $r^{-1}$ such that $$r \cdot r^{-1} = r^{-1} \cdot r = 1$$

Another important example is the group of residues modulo $m$ with respect to modular addition. Recall that its elements can be viewed as the integers from $0$ to $m-1$, and their modular sum is just the remainder of their ordinary sum divided by $m$. We'll denote this group as $(\mathbb{Z}_m, +)$ , and write its elements with a line over: $\overline{0}, \overline{1}, \ldots, \overline{m - 1}$. For instance, $$\mathbb{Z}_5 = \{\overline{0},\overline{1}, \overline{2}, \overline{3}, \overline{4}\}$$The addition in $\mathbb{Z}_m$ is associative, $\overline{0}$ is the neutral element, and for $\overline{x}$, the role of the inverse is played by $\overline{m - x}$ .

Abelian groups

Among the groups discussed above, the group $(S(X), \circ)$ of bijections stands out.

In $(\mathbb{Z}, +)$, $(\mathbb{Z}_m, +)$, or $(\mathbb{R}^{*}, \cdot)$, you can change the order of two operands and the result will stay the same. On the other hand, $$f \circ g \neq g \circ f$$ in the general case of arbitrary $f, g \in S(X)$.

Definition. A group $(G, \star)$ with the additional property that $$a \star b = b \star a \text{ for any } a, b \in G$$ is called commutative, or abelian (after the name of Norwegian mathematician Niels Henrik Abel) group.

The groups $(\mathbb{Z}, +)$, $(\mathbb{Z}_m, +)$, or $(\mathbb{R}^{*}, \cdot)$ are abelian. $(S(X), \circ)$ is non-abelian, and there exist other examples of non-abelian groups. That's why in properties 2 and 3 of a group we have written $$a \star a' = a' \star a = e$$ and $$a \star a' = a' \star a = e,$$mentioned the products in both possible orders.

Conclusion

In this topic, we got acquainted with the concept of a group, seeing the common properties of seemingly different objects. Such generalization allows us not to spend our attention on unnecessary details and thus turns out to be very useful not only for theoretical research but also for practical applications. We have learned that:

• A binary operation takes two elements and produces a third, new element. We denote it like this: $a \star b$, where $\star$ represents the symbol of operation, while $a$ and $b$ are the so-called operands.
• For a set with a binary operation to be called a group has to satisfy 3 conditions: associativity, presence of an identity element, and the existence of an inverse. These 3 conditions must hold true for all the elements in the set.
• Abelian groups are groups that also meet additional criteria in commutativity, i.e. $a \star b = b \star a \text{ for any } a, b \in G$.