Cyclic group:Z4

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This particular group is a finite group of order: 4 This particular group is the smallest (in terms of order): nontrivial non-simple group

Definition

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 $x^l{x}^m=x^{l+m}$ 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

Element	$e$ (identity element)	$x$ (generator)	$x^2$	$x^3$ (generator)
$e$	$e$	$x$	$x^2$	$x^3$
$x$	$x$	$x^2$	$x^3$	$e$
$x^2$	$x^2$	$x^3$	$e$	$x$