Non-isomorphic groups

Dear Dr. Kim,

I am having a few problems with regards to some work on groups. I would like to know how to find isomorphically distinct groups in the context of a semi-direct product, given a number of possible homomorphisms.

Is it enough to show that if the center’s of each possible group are distinct?

If so, how would you find the center of semidirect product with C_10 x C_4 as the underlying set with h:C_4 –>Aut(C_10) a homomorphism, such that h(y) = phi_9, y belongs to C_4, phi_9 belongs to Aut(C_10).

I am aware that the centre is the fixed point set acting on itself by conjugation, however when I try to evaluate this set, denoted Z(g), I can only find that 1, x^5 belong to Z(g) where x^5 belongs to C_10. I have seen elsewhere that the center of this set is isomorphic to C_2 x C_2 but am unable to compute this. Do you see what I have overlooked?

Is this the most efficient way to find distinct groups and eliminate those that are not? I have seen some examples where two groups where found to be the same under a change of variable. Are there any rules or guidelines to take into account when looking for these things, or is it just a matter of remembering them?

Thanks very much for your time.

$Z(g)=\{x\in g: \ \ \forall y\in g,\ \ xy=yx\}$
The general problem of distinguishing groups can eventually involve a great deal of ingenuity and knowledge. For example, if $Z(G_1)$ and $Z(G_2)$ are isomorphic, it can be quite complicated. One good way to get a feel for such problems is via constructing yourself some tricky examples. Can you come up with $G_1$ and $G_2$ that are definitely non-isomorphic, but for which $|G_1|=|G_2|$ and $Z(G_1)\simeq Z(G_2)$?