Element structure of Klein four-group

The Klein four-group is the unique (up to isomorphism) non-cyclic group of order four. In this group, every non-identity element has order two.



The multiplication table with non-identity elements $$a,b,c$$ and identity element $$e$$:

The multiplication table can be described as follows (and this characterizes the group):


 * The product of the identity element and any element is that element itself.
 * The product of any non-identity element with itself is the identity element.
 * The product of two distinct non-identity elements is the third non-identity element.

Elements
Below is a description of the elements for the many alternate descriptions of the Klein four-group. Note that the choice of correspondence between the descriptions is somewhat arbitrary, in the sense that it can be modified by automorphisms of the Klein four-group, which include arbitrary permutations of the three non-identity elements (see endomorphism structure of Klein four-group).

Here is the multiplication table (better termed an addition table, since we are carrying out coordinate-wise addition mod 2) viewed as a direct product of two copies of the additive group of integers mod 2:

Here is the multiplication table viewed as a direct product of two copies of $$\{ -1, 1 \}$$: