Direct Product of Groups - Examples

Examples

  • Let R be the group of real numbers under addition. Then the direct product R × R is the group of all two-component vectors (x, y) under the operation of vector addition:
    (x1, y1) + (x2, y2) = (x1 + x2, y1 + y2).
  • Let G and H be cyclic groups with two elements each:
G
* 1 a
1 1 a
a a 1
H
* 1 b
1 1 b
b b 1
Then the direct product G × H is isomorphic to the Klein four-group:
G × H
* (1, 1) (a, 1) (1, b) (a, b)
(1, 1) (1, 1) (a, 1) (1, b) (a, b)
(a, 1) (a, 1) (1, 1) (a, b) (1, b)
(1, b) (1, b) (a, b) (1, 1) (a, 1)
(a, b) (a, b) (1, b) (a, 1) (1, 1)

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