List of Small Groups - List of Small Abelian Groups

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
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)
Z8 × Z2 Z2 (3), Z4 (2), Z 2
2, Z8 (2), Z4 × Z2
Z 2
4 		Z2 (3), Z4 (6), Z 2
2, Z4 × Z2 (3)

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