Website owner: James Miller
Subgroups of the additive group of integers
The following are subgroups of the additive group of integers ... -3, -2, -1, 0, 1, 2, 3, ...
1. The group generated by positive and negative “powers” of the integer 2 i.e. 2i where the “product” corresponds to addition. Example. 22 = 2 + 2, 23 = 2 + 2 + 2, etc.
.... -6, -4, -2, 0, 2, 4, 6, ....
2. The group generated by 3i
.... -9, -6, -3, 0, 3, 6, 9, ....
3. The group generated by 4i
.... -16, -12, -8, -4, 0, 4, 8, 12, 16, ....
4. The group generated by ai, where i = -∞ to i = ∞
.... -3a, -a, -a, 0, a, 2a, 3a, ....
The above are cyclic groups generated by a single element.
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