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Textbook:

"The following are equivalent for subgroups G1, G2, ...... ,GN of a group.

1) (G1*G2*...*G(K-1)) M GK = {1} for each k=2,3,....,n

2) If g1*g2*....*gn = 1, where each gi is an element of Gi, then gi = 1 for each i."

If these conditions are met then the subgroups are called unconnected.

My question is this: Isn't this just the same as saying that the intersection of any two subgroups is {1}? If not, why? What's the difference?