| We have seen two groups of order 2: Z2 and the
subgroup H1 = {0, 3} of Z6. |
| Though Z2 and H1 are not identical,
they can't be too different: both have only one non-identity element. |
| We shall see that simply having the same number of elements is not
enough to consider two groups equivalent as groups: both must behave in similar fashions
under their group operations. |
| Specifically, we say groups G1 and G2 are
isomorphic if there is a
function f:G1 -> G2 with these properties: |
(i) f is 1-1 and onto (so the elements of G1 and G2 are
in 1-1 correspondence; for finite groups this means they have
the same number of elements), |
(ii) f preserves the group operations, that is, for all
a, b in G1, f(a+b) = f(a) + f(b), where + denotes
the operation in both G1 and G2, and |
(iii) the inverse function f-1 preserves the
group operations. |
| For example, the function f(x) = 3x is an isomorphism
f:Z2 -> H1. |
| We say Z2 is isomorphic to H1, and write
Z2 = H1. |
| Note that
an isomorphism f:G1 -> G2 must take the identity element
of G1 to the identity element of G2. |
| To see a simple example of two non-isomorphic groups having the same number of elements,
consider |
|
Z2 x Z2 =
{(0,0), (1,0), (0,1), (1,1)} |
| It is easy to see that 1 in Z4 has order 4, yet
the elements of
Z2 x Z2 have order 1 or 2. |
| Consequently, Z4 cannot be isomorphic to
Z2 x Z2. |