Problems in group theory, Let i be an integer with 1

Problems in group theory

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Reference no: EM13125617

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*

Reference no: EM13125617

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