# Problems in Group Theory

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*

https://brainmass.com/math/group-theory/problems-group-theory-472744

#### Solution Preview

We are given the group where the are arbitrary groups. We let be the subset of G given by

(a) We wish to show that

First we must show that is a subgroup of G. To see this, we note that the identity of G, namely the element , belongs to , and ...

#### Solution Summary

We solve various problems in group theory pertaining to a given group.