Nine groups are given below Give a complete list of all subgroup relations, of the form G i G J , that exist between these given groups G 1 , G 2 , , G 9 G 1 Z under addition G 2 12Z under addition G 3 Q under multiplication G 4 R under addition G 5 JR under multiplication G 6 n n E Z under multiplication G 7 3Z under addition G 8 the set of all integral multiples of 6 under addition G 9 6 n n E Z under multiplication

Question: Nine groups are given below. Give a complete list of all subgroup relations, of the form G i G J , that exist between

Nine groups are given below. Give a complete list of all subgroup relations, of the form G_i ≤ G_J, that exist between these given groups G₁, G₂, • • •, G₉•

G₁ = Z under addition

G₂= 12Z under addition

G₃ = Q⁺ under multiplication

G₄ = R under addition

G₅= JR+ under multiplication

G₆ = {πⁿ |n E Z} under multiplication

G₇ = 3Z under addition

G₈ = the set of all integral multiples of 6 under addition

G₉ = {6ⁿ |n E Z} under multiplication

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