Understanding the definition of a group

The definition of a group can be puzzling to people encountering it for the first time, particularly if this is the first abstract algebraic definition they're encountering. In fact, it took more than a hundred years (from 1770 to 1880) for mathematicians to move from the intuitive notion of group to the formal definition now employed.

This survey article looks at the definition of a group, its various facets, and how these can be better understood and remembered. We approach this by looking at and addressing some of the common questions and concerns that students have when they first encounter the definition.

Why not commutativity?
If you have dealt with integers, rational numbers, real numbers, and complex numbers, commutativity may seem as natural an assumption to make as associativity, identity elements and inverses. However, you may not have seen, to the same extent, mathematical structures where operations do not commute. So, the fact that the definition of group doesn't include commutativity, may be bothersome.

The key motivation behind dropping commutativity is that most of the interesting groups are not commutative, and that's because most interesting groups arise as symmetries of a structure, and symmetry-rich structures usually have symmetries that do not commute. Loosely, the order in which we do transformations matters.

The simplest example of this is the group of symmetries on three boxes, labeled $$A,B,C$$. Exchanging the labels on boxes labeled $$A$$ and $$B$$, and then exchanging the labels on boxes labeled $$B$$ and $$C$$, has a different effect from first exchanging the labels on boxes labeled $$B$$ and $$C$$, and then exchanging the labels on boxes labeled $$A$$ and $$B$$. Intuitively, the first label exchange changes the backdrop in which the second label exchange occurs.

Noncommutativity in most cases can be traced to a similar cause: the first symmetry changes the backdrop for the second symmetry. This is also more in line with our intuition of change in the real world: preparing for a test and then writing the test has a profoundly different outcome compared to writing the test and then preparing for it. Similarly, chopping a vegetable and then heating it is likely to produce a different result from heating the vegetable and then chopping it. That's because in both cases, one operation changes the backdrop in which the other operation is performed.

Thus, groups are suited to represent situations where things are acting, and when things are acting, the order of operations matters. In contrast, when working with numbers, we're dealing with static quantities.

Why associativity?
Another question that comes up naturally: why the assumption of associativity in the group? Associativity is a natural assumption to make if we are thinking of the group as acting. That's because function composition is associative. To take an analogy with the real world, doing action 1, then (action 2 then action 3), is equivalent to doing (action 1 then action 2), then action 3.

There are certain situations where we have actions that do not satisfy strict assumptions of associativity, but these are more subtle ones. The usefulness of associativity, in general, outweighs the greater generality we get by relaxing it.

Why closure, and identity elements?
A source of angst and confusion for some is the question of why a group should waste an element to be its identity element. If we're thinking of the elements of group as symmetries of a structure, or operations on a structure, it seems wasteful to reserve a place for the operation that, in effect, does nothing: namely the identity element.

The identity element, however, is needed if we need to make sense of the notion of inverting an operation, and if we want our operations to be closed under composition. For instance, if we're looking at the symmetries of a circle, then a rotation by $$\pi$$ is a nontrivial symmetry, but doing such a rotation twice gives the identity operation. If the identity element weren't in the group, we wouldn't have closure.

This leads to the more basic question: why seek a group that is closed under the binary operation?

The question is not a silly one, because in the first few years when groups of transformations were studied, those studying them did not assume the group to be closed under the binary operation or under inverses. However, the developments in the structure theory of groups, for instance, results about the size of the group, the behavior of subgroups, and the nature of elements in the group, heavily relied on the group being closed under the operations. In a sense, the theory of groups cannot be developed properly without making the groups wholesome and closed.

The question of whether we can choose a small collection of elements of the group that generate the group is of independent interest.

Why inverses?
Why should a group have inverses? The idea here is that every operation can be undone -- this has some powerful implications for the structure of the group. For instance, inverses force a cancellation property for elements ($$gh = gk \implies h = k$$), and ensure a uniformity in the structure.

Groups without inverses, called monoids (and semigroups, if we drop the assumption of an identity element), are of independent interest, but their structure theory is nowhere near as elegant and tame as that of groups.

Why not define groups as collections of symmetries?
If groups are being thought of as symmetries of a structure, then why have an abstract definition that talks of a set with a binary operation? This is again a nontrivial question, and it took more than a hundred years since the introduction of the idea of groups of transformations, to settle on the abstract definition of group (given by von Dyck). The key advantage of abstraction is that it allows us to study phenomena that are intrinsic to the group, rather than dependent on the specific way it is acting on a set. It also allows us to think of groups without having a particular action in mind. For instance, we have already studied groups like $$\mathbb{Z}, \mathbb{Q}, \R, \mathbb{C}$$, without thinking of them specifically as acting on sets.

In fact, many of the results in group theory are proved by considering a new action of the group. For instance, the group may act on the set of its elements by left multiplication, it may act on itself as automorphisms by conjugation, it may act on the set of its subgroups by conjugation. It may act on a set, vector space, manifold, or simplicial complex. Each kind of action comes with its own tools of study, and the abstract definition serves as a way of tying all these together.