Verifying the group axioms

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This survey article deals with the question: given a set, and a binary operation, how do we verify that the binary operation gives the set a group structure? This article views the definition of a group as a checklist of conditions.

The general procedure

Define the set and binary operation clearly

First, identify the set clearly; in other words, have a clear criterion such that any element is either in the set or not in the set. For convenience, we'll call the set $G$ .

Second, obtain a clear definition for the binary operation. The binary operation is a map:

$* : G \times G \to G$

In particular, this means that:

$g * h$ is well-defined for any elements $g, h \in G$
The value of $g * h$ is again an element in $G$

Thus, for instance, the operation which sends real numbers $x, y$ to $x^{y}$ is not well-defined when $x$ is negative and $y$ is not an integer; hence, it does not qualify as a binary operation.

Verify associativity

Associativity requires one to pick three arbitrary elements $g, h, k \in G$ , and show that:

$g * (h * k) = (g * h) * k$

There are various strategies for proving this:

If $G$ is a finite set, this may reduce to checking it on all possible triples of elements in $G$
If $*$ is described by means of a mathematical expression, we may be able to simplify the expressions on both sides in terms of variables $g, h, k$ , and show that both sides are equal.
If $G$ is described as a collection of maps from some set $S$ to itself, and the binary operation in $G$ is by composition of maps, then associativity is automatic because function composition is associative

Find an identity element

An identity element (also called neutral element)is an element $e \in G$ such that, for all $g \in G$ :

$g * e = e * g = g$

Again, we have some strategies:

If $G$ is a finite set, this may reduce to checking by inspection.
If $*$ is described by means of a mathematical expression, we may be able to solve a generic equation of the form $g * e = g$
If $G$ is described as a collection of maps from some set $S$ to itself, and the binary operation in $G$ is by composition of maps, the identity element is the identity map

Find an inverse map

Next, we need to demonstrate that for every element $g \in G$ , there exists $h \in G$ such that:

$g * h = h * g = e$

Again, we have some strategies:

If $G$ is a finite set, this may reduce to checking by inspection.
If $*$ is described by means of a mathematical expression, we may be able to solve a generic equation of the form $g * h = e$ for $h$ in terms of $g$
If $G$ is described as a collection of maps from some set $S$ to itself, and the binary operation in $G$ is by composition of maps, the inverse of an element is its inverse as a function

In some special cases

In some special cases, we can by-pass checking various conditions for being a group. We discuss two special cases here:

Subset of a group

Further information: Sufficiency of subgroup criterion

Suppose $G$ is given to be a subset of a group $K$ , and the binary operation on $G$ is the restriction to $G$ of the multiplication in $K$ . Then:

We need to verify that the binary operation induces a well-defined binary operation in $G$ : the product of two elements in $G$ is also in $G$ .
We do not need to check associativity of the binary operation, because it holds in $K$
Instead of trying to find the identity element of $G$ , we can simply verify that the identity element in $K$ , actually lies inside $G$
Instead of trying to compute the inverse map in $G$ , we can simply verify that the inverse map in $K$ , sends $G$ to within itself.

Quotient of a group by an equivalence relation

Suppose $G$ is obtained as the quotient of a group $K$ by an equivalence relation. We want to see whether this equips $G$ with the structure of a group. In this case, the only thing we need to check is that the equivalence relation is a congruence. In other words, if $≃$ is the equivalence relation, we need to check that:

$a ≃ b, c ≃ d ⟹ a c ≃ b d$

Some worked-out examples