Let

G_(1),G_(2),cdots,G_(n)

be a finite collection of groups. The external direct product of these groups, written

G_(1)o+G_(2)o+cdotso+G_(n)

, is

{(g_(1),g_(2),cdots,g_(n)):g_(i)inG_(i)}

where the operation is componentwise, that is,

(g_(1),g_(2),cdots,g_(n))(g_(1)^('),g_(2)^('),cdots,g_(n)^('))=(g_(1)g_(1)^('),g_(2)g_(2)^('),cdots,g_(n)g_(n)^('))

. It is easy to show that this is a group. Use the definition to answer these questions. Is

Z_(2)o+Z_(3)~=S_(3)

? Is

Z_(2)o+Z_(3)~=Z_(6)

?

Let G1,G2,,Gn be a finite collection of groups. The external direct product of these groups, written G1G2Gn, is {(g1,g2,,gn):giGi} where the operation is componentwise, that is, (g1,g2,,gn)(g1,g2,,gn)=(g1g1,g2g2,,gngn). It is easy to show that this is a group. Use the definition to answer these questions. Is Z2Z3S3 ? is Z2Z3Z6