The Klein four-group, commonly denoted as \( V_4 \) or \( K_4 \), is an important concept in group theory. It consists of four elements, which can be represented as:
\[ V_4 = { e, a, b, c } \]
where:
- \( e \) is the identity element,
- \( a \), \( b \), and \( c \) are the other three elements.
The group operation in \( V_4 \) defines the structure of the group:
- \( a \cdot a = e \)
- \( b \cdot b = e \)
- \( c \cdot c = e \)
- \( a \cdot b = c \)
- \( a \cdot c = b \)
- \( b \cdot c = a \)
A useful way to visualize the Klein four-group is through a group table or Cayley table:
\[ \begin{array}{c|cccc} \cdot & e & a & b & c \ \hline e & e & a & b & c \ a & a & e & c & b \ b & b & c & e & a \ c & c & b & a & e \ \end{array} \]
Alternatively, one can represent \( V_4 \) as the group of symmetries of a rectangle, where each of the non-identity elements corresponds to a reflection across the axes of symmetry.
Here is a visual representation of these elements (imagine drawing this):
- Label the corners of a rectangle with \( a \), \( b \), and \( c \).
- The identity element \( e \) can be represented as the rectangle itself remaining unchanged.
In summary, the Klein four-group is a simple group that exemplifies properties like closure, associativity, the existence of identity, and the existence of inverses. It's an abelian group, meaning that the order of the operation does not matter (i.e., \( a \cdot b = b \cdot a \)).