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Dihedral group of order 8
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Dihedral group of order 8
In mathematics, D4 (sometimes alternatively denoted by D8) is the dihedral group of degree 4 and order 8. It is the symmetry group of a square.
As an example, consider a square of a certain thickness with the letter "F" written on it to make the different positions distinguishable. In order to describe its symmetry, one can form the set of all those rigid movements of the square that do not make a visible difference (except the "F"). For instance, if an object turned 90° clockwise still looks the same, the movement is one element of the set, for instance, . One could also flip it around a vertical axis so that its bottom surface becomes its top surface, while the left edge becomes the right edge. Again, after performing this movement, the square looks the same, so this is also an element of our set, which is called it . The movement that does nothing is denoted by .
With composition as the operation, the set of all those movements forms a group. This group is the most concise description of the square's symmetry. Applying two symmetry transformations in succession yields a symmetry transformation. For instance a a, also written as a2, is a 180° degree turn. a3 is a 270° clockwise rotation (or a 90° counter-clockwise rotation). We also see that b2 = e and also a4 = e. A horizontal flip followed by a rotation, a b is the same as b a3. Also, a2 b is a vertical flip and is equal to b a2.
The two elements a and b generate the group, because all of the group's elements can be written as products of powers of a and b.
This group of order 8 has the following Cayley table:
For any two elements in the group, the table records what their composition is. Here we wrote "a3b" as a shorthand for a3 b. This group has 5 conjugacy classes, they are .
In mathematics this group is known as the dihedral group of order 8, and is either denoted Dih4, D4 or D8, depending on the convention. This is an example of a non-abelian group: the operation here is not commutative, which can be seen from the table; the table is not symmetrical about the main diagonal.
There are five different groups of order 8. Three of them are abelian: the cyclic group C8 and the direct products of cyclic groups C4×C2 and C2×C2×C2. The other two, the dihedral group of order 8 and the quaternion group, are not.
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Dihedral group of order 8
In mathematics, D4 (sometimes alternatively denoted by D8) is the dihedral group of degree 4 and order 8. It is the symmetry group of a square.
As an example, consider a square of a certain thickness with the letter "F" written on it to make the different positions distinguishable. In order to describe its symmetry, one can form the set of all those rigid movements of the square that do not make a visible difference (except the "F"). For instance, if an object turned 90° clockwise still looks the same, the movement is one element of the set, for instance, . One could also flip it around a vertical axis so that its bottom surface becomes its top surface, while the left edge becomes the right edge. Again, after performing this movement, the square looks the same, so this is also an element of our set, which is called it . The movement that does nothing is denoted by .
With composition as the operation, the set of all those movements forms a group. This group is the most concise description of the square's symmetry. Applying two symmetry transformations in succession yields a symmetry transformation. For instance a a, also written as a2, is a 180° degree turn. a3 is a 270° clockwise rotation (or a 90° counter-clockwise rotation). We also see that b2 = e and also a4 = e. A horizontal flip followed by a rotation, a b is the same as b a3. Also, a2 b is a vertical flip and is equal to b a2.
The two elements a and b generate the group, because all of the group's elements can be written as products of powers of a and b.
This group of order 8 has the following Cayley table:
For any two elements in the group, the table records what their composition is. Here we wrote "a3b" as a shorthand for a3 b. This group has 5 conjugacy classes, they are .
In mathematics this group is known as the dihedral group of order 8, and is either denoted Dih4, D4 or D8, depending on the convention. This is an example of a non-abelian group: the operation here is not commutative, which can be seen from the table; the table is not symmetrical about the main diagonal.
There are five different groups of order 8. Three of them are abelian: the cyclic group C8 and the direct products of cyclic groups C4×C2 and C2×C2×C2. The other two, the dihedral group of order 8 and the quaternion group, are not.
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