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https://doi.org/10.1007/978-1-4302-5179-8Frieze and wallpaper groups are discovered and studied more than one hundred years ago. In this chapter we study classification of two-dimensional repeated patterns in terms of their respective symmetry groups, the well-known seven frieze groups, and seven wallpaper groups.

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Simple Groups,What are simple groups? Galois called a group simple if its only normal subgroups are the identity subgroup and the group itself.

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Normal Series,To give insights into the structure of a group ., we study a normal series of .. The idea of normal series of a group and solvability yields invariant of groups (the Jordan–Hölder theorem), showing that simple groups are, in certain sense, building towers of finite groups.

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,Symmetry Groups of Geometric Objects,In this chapter we will study interactions between geometry and group theory. The notion of isometry is a general notion commonly accepted in mathematics.

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Platonic Solids,We recall that a polygon is a geometric object consisting of a number of points, vertices, and an equal number of line segments and sides, namely a cyclically ordered set of points in a plane, with no three successive points collinear, together with the line segments joining consecutive pairs of the points.

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