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) |
Read more about this topic: List Of Small Groups
Famous quotes containing the words list of, list, small and/or groups:
“Do your children view themselves as successes or failures? Are they being encouraged to be inquisitive or passive? Are they afraid to challenge authority and to question assumptions? Do they feel comfortable adapting to change? Are they easily discouraged if they cannot arrive at a solution to a problem? The answers to those questions will give you a better appraisal of their education than any list of courses, grades, or test scores.”
—Lawrence Kutner (20th century)
“Thirtythe promise of a decade of loneliness, a thinning list of single men to know, a thinning brief-case of enthusiasm, thinning hair.”
—F. Scott Fitzgerald (18961940)
“To make us feel small in the right way is a function of art; men can only make us feel small in the wrong way.”
—E.M. (Edward Morgan)
“Instead of seeing society as a collection of clearly defined interest groups, society must be reconceptualized as a complex network of groups of interacting individuals whose membership and communication patterns are seldom confined to one such group alone.”
—Diana Crane (b. 1933)