Are two finite groups of the same order always isomorphic? Are two finite groups of the same order always isomorphic? Some simple example would be great!
 A: Not at all...$\mathbb{Z}_4$ and the Klein's 4-group $K_4$ both are finite groups of order 4 but not isomorphic.
Many other examples are there. There are two non-isomorphic groups of order 6, viz. $\mathbb{Z}_6$ and $S_3$.
A: Just an added note: 
All finite groups of prime order $p$ are isomorphic, and they are all isomorphic to $\langle \mathbb Z_p, +_p\rangle$, where $+_p$ is addition modulo p.
This makes a nice exercise to prove it. I'm sure you can search math.stackexchange to check your proof and compare with others.
A: I don't know if you know the group action but, you can review this counterexample for the futhure. Consider the following groups:
$$G_1=\{\text{id},(1,2),(3,4),(1,2)(3,4)\}\\ G_2=\{\text{id},(1,3)(2,4),(1,4)(2,3),(1,2)(3,4)\}$$
Although, both are isomorphic to $\mathbf{V}$ of order $4$; they are not isomorphic of permutation groups point of view. In fact, it easy to check that $(G_2\mid\Omega)$ transitively while $(G_1\mid\Omega)$ is not wherein $\Omega=\{1,2,3,4\}$.
A: There are 


*

*56 092 pairwise non-isomorphic groups of order 256

*10 494 213 pairwise non-isomorphic groups of order 512

*49 487 365 422 pairwise non-isomorphic groups of order 1024

A: The smallest counterexample are the groups of order $4$.
The Klein four group $\mathbb Z/2\mathbb Z\times \mathbb Z/2\mathbb Z$ and the cyclic group $\mathbb Z/4\mathbb Z$ both have order $4$. However, they are not isomorphic, since only the latter one contains an element of order $4$.
Addition:
In the representation $\mathbb Z/2\mathbb Z\times \mathbb Z/2\mathbb Z$, the Klein four group is the set
$$ \{(0,0),\quad (1,0),\quad (0,1),\quad (1,1)\}$$
together with the entry-wise addition mod $2$.
Another quite common representation of the Klein four group is the subgroup of the symmetric group $S_4$ consisting of the identity and the $3$ double transpositions:
$$
\{\operatorname{id},\quad (12)(34),\quad (13)(24),\quad (14)(23)\}
$$
A quite concrete representation of the Klein four group is given by the symmetry group of a rectangle.
A: $$\mathbb{Z}_6\not\equiv S_3$$
http://www.google.lk/url?url=http://en.wikipedia.org/wiki/List_of_small_groups&rct=j&frm=1&q=&esrc=s&sa=U&ei=tIhQVKrGH5Ld8AXG6oKoDg&ved=0CBIQFjAA&usg=AFQjCNEDmUxE-_sIsv8ctvlylO9YnpofQg
A: As the others said already. The answer is: No.
An easy to find examples is just to find a group which is not cyclic. Say you found a group $G_1$ of order $m$ which is not cyclic. Consider now the group $G_2=(\mathbb Z/m\mathbb Z,+)$. Since $G_2$ is cyclic with generator $\overline{1}$ you have found an example.
It does worth to note that two finite groups with the same prime order are isomorphic. 
In order to see this it is sufficient to show that any group of prime order is cyclic. (Take an arbitrary element and look at the generated group). Then define a homomorphism which sends the generator of the first group to the generator of the second group.
