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:

When the binary operation is commutative

When * is commutative, then it suffices to find a left identity element, and it suffices to compute just a left inverse (or just a right inverse).

Subset of a group

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 \sim is the equivalence relation, we need to check that:

a \sim b, c \sim d \implies ac \sim bd

Some worked-out examples

An abelian group

Here is one example. Consider G = \mathbb{R} \setminus \{ -1 \} and define, for x,y \in G:

x * y := x + y + xy

We want to show that (G,*) is a group.

First, we check the closure of G under *. Namely, we need to check that if x,y \in G then x * y \in G. Suppose not. Then, we have:

x + y + xy = - 1 \implies (x+1)(y+1) = 0

which would force either x = -1 or y = -1, a contradiction to x,y \in G.

Next, we need to check associativity. We do this using the generic formula. We get:

(x * y) * z = (x + y + xy) + z + (x + y + xy)z = x + y + z + xy + yz + xz + xyz

and we also have:

x * (y * z) = x + (y + z + yz) + x(y + z + yz) = x + y + z + xy + yz + xz + xyz

Now, observe that * is commutative (it is symmetric in x and y). So it suffices to compute a one-sided identity element and verify the existence of one-sided inverses.

First, we need to find the identity element. In other words, for any x \in G, we want:

x * e = x \implies x + e + xe = x \implies e (1 + x) = 0

Since x \ne -1, we get e = 0.

Finally, we need to compute the inverse map:

x * y = 0 \iff x + y + xy = 0 \implies y = -\frac{x}{1 + x}

This gives a formula for the inverse map. Note first that the formula makes sense, because x \ne -1, so 1 + x \ne 0. Further, the output is not -1, because solving -1 = -\frac{x}{1 + x} gives 1 = 0, a contradiction. The inverse map is thus a well defined map from G to G.

Thus, (G,*) is a group with identity element 0 and inverse map:

x \mapsto \frac{-x}{1 + x}

A group of symmetries

Here's another example. Suppose S is a finite set of points in \mathbb{R}^3. Suppose G is the set of all maps f: S \to S such that for any x,y \in S, the distance between f(x) and f(y) equals the distance between x and y. Define a binary operation in G by composition:

(f * g)(x) = (f \circ g)(x) = f(g(x))

We want to show that (G,*) is a group. Note that G is realized as a set of functions under composition.

  • Closure of G under * follows from the transitivity of the relation of distances being equal.
  • Associativity follows from the fact that function composition is associative. Explicitly:

(f * (g * h))(x) = f((g * h)(x)) = f(g(h(x)))

and similarly:

((f * g) * h)(x) = (f * g)(h(x)) = f(g(h(x)))

Since this equality holds for every x \in S, we have:

f * (g * h) = (f * g) * h

  • The identity element is the identity map from S to S. This clearly satisfies the condition for being an element of G.
  • To show that every map has an inverse, we first observe that any f:S \to S that preserves distances must be injective. That's because if f(x) = f(y), then the distance between f(x) and f(y) is zero, so the distance between x and y is zero, so x = y. Since S is a finite set, f must be bijective, so it has a unique inverse map. It is clear that this inverse map also preserves distances, so is in G.