List of Small Abelian Groups
The finite abelian groups are either cyclic groups, or direct products thereof; see abelian groups.
Order | Group | Subgroups | Properties | Cycle graph |
---|---|---|---|---|
1 | trivial group, Z1 = S1 = A2 | - | various properties hold trivially | |
2 | Z2 = S2 = Dih1 | - | simple, the smallest non-trivial group | |
3 | Z3 = A3 | - | simple | |
4 | Z4 | Z2 | ||
Klein four-group, Z 2 2 = Dih2 |
Z2 (3) | the smallest non-cyclic group | ||
5 | Z5 | - | simple | |
6 | Z6 = Z3 × Z2 | Z3, Z2 | ||
7 | Z7 | - | simple | |
8 | Z8 | Z4, Z2 | ||
Z4 × Z2 | Z 2 2, Z4 (2), Z2 (3) |
|||
Z 3 2 |
Z 2 2 (7), Z2 (7) |
the non-identity elements correspond to the points in the Fano plane, the Z2 × Z2 subgroups to the lines | ||
9 | Z9 | Z3 | ||
Z 2 3 |
Z3 (4) | |||
10 | Z10 = Z5 × Z2 | Z5, Z2 | ||
11 | Z11 | - | simple | |
12 | Z12 = Z4 × Z3 | Z6, Z4, Z3, Z2 | ||
Z6 × Z2 = Z3 × Z 2 2 |
Z6 (3), Z3, Z2 (3), Z 2 2 |
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13 | Z13 | - | simple | |
14 | Z14 = Z7 × Z2 | Z7, Z2 | ||
15 | Z15 = Z5 × Z3 | Z5, Z3 | multiplication of nimbers 1,...,15 | |
16 | Z16 | Z8, Z4, Z2 | ||
Z 4 2 |
Z2 (15), Z 2 2 (35), Z 3 2 (15) |
addition of nimbers 0,...,15 | ||
Z4 × Z 2 2 |
Z2 (7), Z4 (4), Z 2 2 (7), Z 3 2, Z4 × Z2 (6) |
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Z8 × Z2 | Z2 (3), Z4 (2), Z 2 2, Z8 (2), Z4 × Z2 |
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Z 2 4 |
Z2 (3), Z4 (6), Z 2 2, Z4 × Z2 (3) |
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