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:
“A mans interest in a single bluebird is worth more than a complete but dry list of the fauna and flora of a town.”
—Henry David Thoreau (18171862)
“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)
“The ancestral deed is thought and done,
And in a million Edens fall
A million Adams drowned in darkness,
For small is great and great is small,
And a blind seed all.”
—Edwin Muir (18871959)
“In America every woman has her set of girl-friends; some are cousins, the rest are gained at school. These form a permanent committee who sit on each others affairs, who come out together, marry and divorce together, and who end as those groups of bustling, heartless well-informed club-women who govern society. Against them the Couple of Ehepaar is helpless and Man in their eyes but a biological interlude.”
—Cyril Connolly (19031974)