Isomorphisms: Abstract Algebra?

Question

How do you prove that the quaternion group is not isomorphic to the dihedral group D_4?

the quanternion group looks like this
.........| e a a^2 a^3 b ba ba^2 ba^3
----------------------------- -------------------
e.......| e a a^2 a^3 b ba ba^2 ba^3
a.......| a a^2 a^3 e ba^3 b ba ba^2
a^2...| a^2 a^3 e a ba^2 ba^3 b ba
a^3...| a^3 e a a^2 ba ba^2 ba^3 b
b.......| b ba ba^2 ba^3 a^2 a^3 e a
ba.....| ba ba^2 ba^3 b a a^2 b^3 e
ba^2.| ba^2 ba^3 b ba e a a^2 a^3
ba^3.| ba^3 b ba ba^2 a^3 e a a^2

Answer 1

Here's one way: list the cyclic subgroups of each group. If the two groups have different subgroup structures, then they are not isomorphic. That is, if the number of cyclic subroups of a certain order is different in one group than in the other, the two groups cannot be isomorphic.

Answer 2

just inspecting your table it looks like the generator b cannot
be the same as the generator in D4 which is order 2