Understanding S(3) and Its Structure
S(3) is the symmetric group on three symbols, representing all possible permutations of three elements. It has 6 elements, which are the different ways to arrange three objects. The group can be expressed as:
- Identity: (1)
- Transpositions: (1 2), (1 3), (2 3)
- 3-cycles: (1 2 3), (1 3 2)
Each element corresponds to a distinct permutation of the set {1, 2, 3}.
Exploring S(3) x Z/2Z
The notation S(3) x Z/2Z represents the direct product of S(3) and the cyclic group of order 2. The group Z/2Z consists of two elements: {0, 1}, where 0 is the identity, and 1 represents a non-trivial element.
The direct product combines both groups, resulting in each element of S(3) being paired with each element of Z/2Z. Thus, the elements of S(3) x Z/2Z can be expressed as:
- (π, 0) for all π in S(3)
- (π, 1) for all π in S(3)
This results in a total of 12 elements (6 from S(3) and 2 from Z/2Z).
Isomorphism to D_3
The group S(3) x Z/2Z is isomorphic to D_3, the dihedral group of order 6. D_3 represents the symmetries of an equilateral triangle and consists of:
- 3 rotations: 0°, 120°, and 240°
- 3 reflections: across the axes of symmetry
The correspondence between elements of S(3) x Z/2Z and D_3 preserves the group operation, establishing that:
- S(3) x Z/2Z ≅ D_3
In conclusion, the group S(3) x Z/2Z is isomorphic to the dihedral group D_3, encapsulating both permutation and symmetry properties.