Cyclic group:Z4

Verbal definition
The cyclic group of order 4 is defined as a group with four elements $$e = x^0, x^1, x^2, x^3$$ where $$x^lx^m = x^{l+m}$$ where the exponent is reduced modulo $$4$$. In other words, it is the cyclic group whose order is four. It can also be viewed as:


 * The quotient group of the group of integers by the subgroup comprising multiples of $$4$$.
 * The multiplicative subgroup of the nonzero complex numbers under multiplication, generated by $$i$$ (a squareroot of $$-1$$).
 * The group of rotational symmetries of the square.

Multiplication table
This is the multiplication table using multiplicative notation:

This is the multiplication table using additive notation, i.e., thinking of the group as the group of integers modulo 4:

Other descriptions
The group can also be defined using GAP's CyclicGroup function as:

CyclicGroup(4)

Internal links

 * Linear representation theory of cyclic group:Z4
 * Group cohomology of cyclic group:Z4
 * Galois extensions for cyclic group:Z4