phukraut

think of a group G with the following operator *.

say x is in G. now consider how x operates with itself, this is a general definition of integer exponents when talking about groups:

x * x * x * ... * x = x^k, for some positive integer k.

now every group has an identity element, call it I. if we try and find k such that, for any y in G,

x^k * y = y, then x^k = I by definition for some k. if k>0, then x^k * y may not be y, so x^0 = I. if G is the normal multiplicative group and * is just multiplication, you have that x^0 = 1.

here's another way to think of it, say we want to find the inverse of x^k. by group theory, for some z in G, x^k * z = I. the answer is z = x^(-k). what if k=0? then for some t, x^0 * x^t = x^(0+t) = I. but 0 is the additive identity, and so by definition, 0+t=0 only if t=0. therefore x^0 is its own inverse. therefore x^0=I.

it's just a useful convention for identity transformations. hope this sheds some light.