In an inverse limit of groups, how can we define group law?

Question

Roughly speaking, if g,h in lim (<--) G_i
g=(g_k) k in K
and h=(h_ j) j in J.

So g+h = ??? we cannot use component-wise law since components don't belong to the same G_i. So I can't understand what you say...I'm sorry.... J and K are not the same set and their intesection can be empty!! I'm probably idiot but I really can't see how to make a group law.

Answer 1

In calculating g+h each element comes from one of the groups, say g from Gi and h from Gj. Then you have a map which injects one group into the next, and so then g in Gi can be thought of as a g-hat in Gj (or vice versa, which ever comes first in the chain, Gi or Gj) and so then you use the group law within that group. In this sense we say the group law is "inherited" or the "natural" group law.