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I am just starting to investigate group theory and wonder why it is seen as so important in the mathematical world. Certainly it seems that it is very flexible in its uses and we continually hear of Galois theory, but what else is there?

Please reply in a fashion that can be understood by someone who, although knowing the basics of group theory, does not know of any particularly important groups (but would be very interested in learning of them)

Thank you for your help!

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    $\begingroup$ As a physicist, I think Noether's theorem, which relates symmetry groups to physically conserved quantities, is pretty important. $\endgroup$ – user137731 Dec 8 '14 at 15:04
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One of the major reasons why groups are important is because groups act on things. You may have seen groups acting on a set. This arises naturally because all groups can be thought of as subgroups of a permutation group. In geometry and topology, groups often act on spaces in a way such that certain structure is preserved. For example, Lie groups act on manifolds, algebraic groups act on algebraic varieties, topological groups act on topological spaces. The symmetries of a space form a group. In general, the automorphisms of any object form a group. You may want to read a bit about homogeneous spaces.

Often when groups act on an object, you can learn a lot about the object itself from studying the group action. Also, the orbit space often has interesting structure. A very basic example of this would be the torus as the orbit space of $\mathbb{R}^2/\mathbb{Z}^2$.

Another reason groups are important in geometry and topology is because they are invariants of spaces. A topological invariant of a topological space is a property which depends only on the topology. For example compactness, or connectedness. There is a class of algebraic invariants, called the homotopy groups, which are very important. The most important homotopy group is called the fundamental group. If two spaces have different fundamental groups, then they are not homeomorphic.

As far as the most important groups go, you probably already know the groups which are arguably the most important: the symmetric group and the general linear group.

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There are many other structures that are defined in terms of groups, e.g. rings, fields, vector spaces. Having groups makes these much neater and some of behaviour of more complex structures is inherited from behaviour of groups (groups in general or special cases, like Abelian groups).

Think of groups as a simple structure appearing within many other structures, and sometimes that's all those structures have in common.

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