List of Small Non-abelian Groups
| Order | Group | Subgroups | Properties | Cycle Graph |
|---|---|---|---|---|
| 6 | S3 = Dih3 | Z3, Z2 (3) | the smallest non-abelian group | |
| 8 | Dih4 | Z4, Z22 (2), Z2 (5) | ||
| quaternion group, Q8 = Dic2 | Z4 (3), Z2 | the smallest Hamiltonian group; smallest group demonstrating that all subgroups may be normal without the group being abelian; the smallest group G demonstrating that for a normal subgroup H the quotient group G/H need not be isomorphic to a subgroup of G | ||
| 10 | Dih5 | Z5, Z2 (5) | ||
| 12 | Dih6 = Dih3 × Z2 | Z6, Dih3 (2), Z22 (3), Z3, Z2 (7) | ||
| A4 | Z22, Z3 (4), Z2 (3) | smallest group demonstrating that a group need not have a subgroup of every order that divides the group's order: no subgroup of order 6 (See Lagrange's theorem and the Sylow theorems.) | ||
| Dic3 = Z3 Z4 | Z2, Z3, Z4 (3), Z6 | |||
| 14 | Dih7 | Z7, Z2 (7) | ||
| 16 | Dih8 | Z8, Dih4 (2), Z22 (4), Z4, Z2 (9) | ||
| Dih4 × Z2 | Dih4 (2), Z4 × Z2, Z23 (2), Z22 (11), Z4 (2), Z2 (11) | |||
| generalized quaternion group, Q16 = Dic4 | ||||
| Q8 × Z2 | Hamiltonian | |||
| The order 16 quasidihedral group | ||||
| The order 16 modular group | ||||
| Z4 Z4 | ||||
| The group generated by the Pauli matrices | ||||
| G4,4 = Z22 Z4 |
Read more about this topic: List Of Small Groups
Famous quotes containing the words list of, list, small and/or groups:
“The advice of their elders to young men is very apt to be as unreal as a list of the hundred best books.”
—Oliver Wendell Holmes, Jr. (1841–1935)
“Every morning I woke in dread, waiting for the day nurse to go on her rounds and announce from the list of names in her hand whether or not I was for shock treatment, the new and fashionable means of quieting people and of making them realize that orders are to be obeyed and floors are to be polished without anyone protesting and faces are to be made to be fixed into smiles and weeping is a crime.”
—Janet Frame (b. 1924)
“When we consider the vast distance of the known and visible parts of the world, and the reasons we have to think, that what lies within our ken is but a small part of the universe, we shall then discover an huge abyss of ignorance.”
—John Locke (1632–1704)
“As in political revolutions, so in paradigm choice—there is no standard higher than the assent of the relevant community. To discover how scientific revolutions are effected, we shall therefore have to examine not only the impact of nature and of logic, but also the techniques of persuasive argumentation effective within the quite special groups that constitute the community of scientists.”
—Thomas S. Kuhn (b. 1922)