The product of i (a positive integer)
copies of the element g in a group is denoted by 
.
This is defined inductively as 
and for 
, 
.
For i>0 we define 
. This is also equal to 
.
It is easily shown that the rules of exponents
, 
hold in a group.
What is not necessarily true is the law of exponents 
which we
use in algebra of the real numbers. This holds in the reals because
multiplication is commutative. Indeed, this law of exponents holds in an
abelian group.
When we use + for the binary operation, these rules translate to 0g = 0 and ig = g + (i-1)g for the inductive definition. Also, (-i)g = i(-g). We have (ig) + (jg) = (i+j)g and j(ig) = (ji)g.