Image Gallery: Group (mathematics)

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The manipulations of the Rubik's Cube form the Rubik's Cube group.
A periodic wallpaper pattern gives rise to a wallpaper group.
The fundamental group of a plane minus a point (bold) consists of loops around the missing point. This group is isomorphic to the integers under addition.
The hours on a clock form a group that uses addition modulo 12. Here, 9 + 4 ≡ 1.
The 6th complex roots of unity form a cyclic group. is a primitive element, but is not, because the odd powers of are not a power of ⁠⁠.
The (2,3,7) triangle group, a hyperbolic reflection group, acts on this tiling of the hyperbolic plane
Two vectors (the left illustration) multiplied by matrices (the middle and right illustrations). The middle illustration represents a clockwise rotation by 90°, while the right-most one stretches the ⁠⁠-coordinate by factor 2.
The unit circle in the complex plane under complex multiplication is a Lie group and, therefore, a topological group. It is topological since complex multiplication and division are continuous. It is a manifold and thus a Lie group, because every small piece, such as the red arc in the figure, looks like a part of the real line (shown at the bottom).

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