Group theory is often introduced a bit imprecisely. Perhaps this is because its practitioners are too familiar and comfortable with it. I will introduce a concise definition of a group and will prove a few important consequences of the definition.
A group is a set of elements, G, with a mapping G x G > G (i.e. all pairs of elements in G are mapped to an element of G). The mapping of an element pair (a,b) to an element c is indicated by the expression ab = c. The mapping, called the group product, must obey the following rules:
 a(bc) = (ab)c (associative property)
 An element e exists such that for all a in G, ea = a.
 For each element a in G, an element b exists such that ba = e. b is denoted by a^{1}.
It is important to realize that ab is not necessarily equal to ba. If ab = ba for all elements a and b, then the group is termed abelian.
That's all! This definition of a group is a bit more concise than the earlier definitions. The extras in those definitions can be proven as consequences of this definition.
Consequences of this definition:
 ab = ac implies b = c.
Proof:
a^{1}(ab) = a^{1}(ac)
(a^{1}a)b = (a^{1}a)c
b = c
 ae = a
Proof:
let ae = d
a^{1}ae = a^{1}d
a^{1}d = e
by definition, a^{1}a = e
from consequence 1, d = a
 e is unique
Proof:
suppose fa = a
f = f
f = fe (because of consequence 2)
f = e
or suppose af = a
f = f
f = ef
f = e
 aa^{1} = e
Proof:
let aa^{1} = d
a^{1}aa^{1} = a^{1}d
a^{1} = a^{1}d
from consequence 2, a^{1}e = a^{1}
from consequence 1, d = e
 ba = ca implies b = c (addendum to consequence 1)
Proof:
baa^{1} = caa^{1}
from consequence 4, b = c
An interesting example of a group is the 4group (Viergruppe) V defined with the following group product. This group is abelian (the group product is commutative). Actually, all groups of order 5 or smaller are abelian.
e a b c

e  e a b c

a  a e c b

b  b c e a

c  c b a e
One could imagine that a represents 180degree rotation about the xaxis, b represents 180degree rotation about the yaxis, c represents 180degree rotation about the zaxis, e represents no rotation, and the group product means that one operation is followed by another. You can perform these rotations with a book in your hand to visualize them. Rotations about different axes are in general not commutative. You can verify this by performing 90degree rotations with the book.