Group theory is the study of the algebraic structures known as “groups”. A group \(G\) is a collection of elements \(X\) paired with an operator \(\bullet\) that combines elements of \(X\) while obeying certain laws. Roughly speaking, \(\bullet\) treats elements of \(X\) as composable, invertible actions.

Group theory has many applications. Historically, groups first appeared in mathematics as groups of “substitutions” of mathematical functions; for example, the group of integers \(\mathbb{Z}\) acts on the set of functions \(f : \mathbb{R} \to \mathbb{R}\) via the substitution \(n : f(x) \mapsto f(x - n)\), which corresponds to translating the graph of \(f\) \(n\) units to the right. The functions which are invariant under this group action are precisely the functions which are periodic with period \(1\), and group theory can be used to explain how this observation leads to the expansion of such a function as a Fourier series \(f(x) = \sum \left( a_n \cos 2 \pi n x + b_n \sin 2 \pi n x \right)\).

Groups are used as a building block in the formalization of many other mathematical structures, including fields, vector spaces, and integers. Group theory has various applications to physics. For a list of example groups, see the examples page. For a list of the key theorems in group theory, see the main theorems page.

Interpretations, visualizations, and uses

Group theory abstracts away from the elements in the underlying set \(X\): the group axioms do not care about what sort of thing is in \(X\); they care only about the way that \(\bullet\) relates them.\(X\) tells us how many elements \(G\) has; all other information resides in the choices that \(\bullet\) makes to combine two group elements into a third group element. Thus, group theory can be seen as the study of possible relationships between objects that obey the group laws; regardless of what the objects themselves are. To visualize a group, we need only visualize the way that the elements relate to each other via \(\bullet\). This is the approach taken by group multiplication tables and group diagrams.

Group theory is interesting in part because the constraints on \(\bullet\) are at a “sweet spot” between “too lax” and “too restrictive.” Group structure crops up in many areas of physics and mathematics, but the group axioms are still restrictive enough to make groups fairly easy to work with and reason about. For example, if the order of \(G\) is prime then there is only one possible group that \(G\) can be (up to isomorphism). There are only 2, 2, 5, 2, and 2 groups of order 4, 6, 8, 9, and 10 (respectively). There are only 16 groups of order 100. If a group structure can be found in an object, this makes the behavior of the object fairly easy to analyze (especially if the order of the group is small). Group structure is relatively common in math and physics; for example, the solutions to a polynomial equation are acted on by a group called the Galois group (a fact from which the unsolvability of quintic polynomials by radicals was proven). Group theory is thus a useful tool for figuring out how various mathematical and physical objects behave. For more on this idea, see the page on group actions.

Roughly speaking, the constraints on \(\bullet\) can be interpreted as saying that \(\bullet\) has to treat the elements like they are a “complete set of transformations” of a physical object. The axiom of identity then says “one of the elements has to be interpreted as a transformation that does nothing”; the axiom of inversion says “each transformation must be invertible;” and so on. In this light, group theory can be seen as the study of possible complete sets of transformations. Because the set of possible groups is relatively limited and well-behaved, group theory is a useful tool for studying the relationship between transformations of physical objects. For more discussion of this interpretation of group theory, see groups and transformations.

Groups can also be seen as a tool for studying symmetry. Roughly speaking, given a set \(X\) and some redundant structure atop \(X\), a symmetry of \(X\) is a function from \(X\) to itself that preserves the given structure. For example, a physicist might build a model of two planets interacting in space, where an arbitrary point picked out as the origin. The behavior of the planets in relation to each other shouldn’t depend on which point they label \((0, 0, 0)\), but many questions (such as “where is the planet now?”) depend on the choice of origin. To study facts that are true regardless of where the origin is (rather than being artifacts of the representation), they might let \(X\) be the possible states of the model, with then ask what properties of \(X\) are preserved when the origin is shifted — i.e., they ask “what facts about this model would still be true if I had picked a different origin?” Translation of the origin is a “symmetry” of those properties: It takes one state of \(X\) to another state of \(X\), and preserves all properties of the physical system that are independent of the choice of origin. When a model has multiple symmetries (such as rotation invariance as well as translation invariance), those symmetries form a group under composition. Group theory tells us about how those symmetries must relate to one another. For more on this topic, see groups and symmetries.

In fact, laws of physics can be interpreted as “properties of the universe that are true at all places and all times, regardless of how you label things.” Thus, the laws of physics can be studied by considering the properties of the universe that are preserved under certain transformations (such as changing labels). Group theory puts constraints on how these symmetries in the laws of physics relate to one another, and thus helps constrain the search for possible laws of physics. For more on this topic, see groups and physics.

Finally, in a different vein, groups can be seen as a building block for other mathematical concepts. As an example, consider numbers. “Number” is a fairly fuzzy concept; the term includes the integers \(\mathbb Z\), the rational numbers \(\mathbb Q\) (which include fractions), the real numbers \(\mathbb R\) (which include numbers such as \(\sqrt{2}\)), and so on. The arrow from \(\mathbb Z \to \mathbb Q \to \mathbb R\) points in the direction of specialization: Each step adds constraints (such as “now you must include fractions”) and thus narrows the scope of study (there are fewer things in the world that act like real numbers than there are that act like integers). Group theory is the result of following the arrow in the opposite direction: groups are like integers but more permissive. Groups have an identity element that corresponds to \(0\) and an operation that corresponds to \(+\), but they don’t need to be infinitely large and they don’t need to have anything corresponding to multiplication. From this point of view, “group” is a generalization of “integer”, and from that generalization, we can build up many different mathematical objects, including rings, fields, and vector spaces. Empirically, many interesting and useful mathematical objects use groups as one of their building blocks, and thus, some familiarity with group theory is helpful when learning about other mathematical structures. For more on this topic, see the tree of algebraic structures.