A thorough explanation of what a binary operation is can be found at Notes on Binary Operations, from the Mathematics Department of California State University San Bernardino. See their Reference Notes Page.

An example of a binary operation (called "!"), on {a, b}.          From Notes on Binary Operations. Used with permission.

See Associative operations, Commutative operations, and Distributive properties, from the Mathematics Department of California State University San Bernardino.

See Identities and Inverse, from the Mathematics Department of California State University San Bernardino.

Groups, from the Mathematics Department of California State University San Bernardino, deals with many topics on the IB syllabus.

Formal definitions of groups, cyclic groups, permutation groups, and more are provided at Groups, part of the Abstract Algebra On Line site.

Also see Groups, from David Reid, Assistant Professor of Mathematics Education at Memorial University of Newfoundland.

For a graduated introduction to groups, see Simple Algebraic Structures. Definitions are given for algebraic structures (groupoids, semigroups, monoids, loops, etc.) that are simpler than groups.

Finally, Exploring Abstract Algebra with Mathematica is an attempt to provide functionality
for working with structures in abstract algebra through the use of packages written for Mathematica. EAAM has several labs available for downloading at Labs for Group Theory. Recommended for users of Mathematica.

An example of an abelian group. The set of elements consists of those non-negative integers less than 14 that are relatively prime to 14. The operation is multiplication modulo 14. From EAAM. Used with permission.

The Groups15 Demo, from John J. Wavrik of the University of California San Diego, is a Java applet allowing experimentation with group tables for groups of orders 1 through 15.

For symmetry groups, see Groups and Symmetry, from the Geometry Forum. Also see Symmetry in the Plane, from David Reid, and Symmetry Groups, from the Mathematics Department of California State University San Bernardino. Symmetry and Group Theory, from The Geometry Junkyard, has a number of examples involving both symmetry and groups. Finally, Symmetry Web, from Oklahoma State University, provides an exploration of the symmetries of such figures as the equilateral triangle, cube and dodecahedron. Exercises and projects are included.

Abstract Algebra On Line has a Permutation Groups entry. Also see Permutation Groups, from the Mathematics Department of CSUSB.

Diagram showing a rotation.
If the diagram is not animated, try reloading.
From Transformations. Used with the permission of David Reid,
Memorial University of Newfoundland.

More many high-level examples, see ATLAS of Finite Group representations. As a sampler, exceptional group E7(2) is of order 7997476042075799759100487262680802918400.

For information on order, see Order of a group, element, from CSUSB's Mathematics Department. The order of a group and the order of an element are both discussed.

Also see Cyclic Groups, from the Mathematics Department of California State University San Bernardino.

The generation of a cyclic subgroup. The operation is multiplication modulo 19. The set of elements generated consists of those non-negative integers less than 19 that are relatively prime to 19. Note that the red dots get progressively larger as the numbers are generated. From EAAM. Used with permission.

See Subgroups, from the Mathematics Department of California State University San Bernardo. Lagrange's theorem is discussed.

Also see Groups - scroll down. Lagrange's theorem is stated (at 3.2.10), as are several corollaries. From Abstract Algebra On Line.

See Isomorphism, from the Mathematics Department of California State University San Bernardino.