Assignment:
Let i be an integer with 1 <= i <= n. Let Gi* be the subset of G1 X ... X Gn consisting of those elements whose ith coorinate is any element of Gi and whose other coordinates are each of the identity element, that is,
Gi* = {(ei,...ei-1,ai,ei+1,...,en | ai in G}
Show that
Gi* is a normal subgroup of G1 X ... X Gn
Gi* is isomorphic to Gi
Gi X ... X Gn is the (internal) direct product of its subgroups G1* , ... , Gn*
(Show that every element G1 X ... X Gn can be written uniquely in the form a1a2...an with ai in Gi*
Provide complete and step by step solution for the question and show calculations and use formulas.