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    Problems in Group Theory

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    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*

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    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.