| Mathematics -- Groups of Small Order |
From http://www.math.usf.edu/~eclark/algctlg/small_groups.html
Groups of small order
Compiled by John Pedersen, Dept of Mathematics, University of South Florida, jfp followed by an at-sign, then math.usf.edu
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Order 1 and all prime orders (1 group: 1 abelian, 0 nonabelian)
- All groups of prime order p are isomorphic to C_p, the cyclic group of
order p.
A concrete realization of this group is Z_p, the integers under addition modulo p. -
Order 4 (2 groups: 2 abelian, 0 nonabelian)
 | C_4, the cyclic group of order 4
| V = C_2 x C_2 (the Klein four group) = symmetries of a rectangle. A
presentation for the group is
| <a, b; a^2 = b^2 = (ab)^2 = 1>The Cayley table of the group is (putting c = ab): | 1 a b c
--+-----------
1 | 1 a b c
a | a 1 c b
b | b c 1 a
c | c b a 1
A matrix representation is the four 2x2 matrices
[1 0] [1 0] [-1 0] [-1 0]
[0 1], [0 -1], [ 0 1], [ 0 -1]
A permutation representation is the following four elements of S_4:
(1), (1 2)(3 4), (1 3)(2 4) and (1 4)(2 3).Its lattice of subgroups is (in the notation of the Cayley table) V
/ | \
<a> <b> <c>
\ | /
{1}
|
Order 6 (2 groups: 1 abelian, 1 nonabelian)
| C_6
| S_3, the symmetric group of degree 3 = all permutations on three
objects, under composition. In cycle notation for permutations, its
elements are (1), (1 2), (1 3), (2, 3), (1 2 3) and (1 3 2). | There are four proper subgroups of S_3; they are all cyclic. There are the three of order 2 generated by (1 2), (1 3) and (2 3), and the one of order 3 generated by (1 2 3). Only the one of order 3 is normal in S_3. A presentation for S_3 is (where s corresponds to (1 2) and t to (2 3)): <s,t; s^2 = t^2 = 1, sts = tst>Another presentation (with s <-> (1 2 3), t <-> (1 2)) is <s,t; s^3 = t^2 = 1, ts = s^2 t>In terms of this second presentation, with 2 = s^2, u = ts and v = ts^2, the Cayley table is | 1 s 2 t u v
--+-----------------------
1 | 1 s 2 t u v
s | s 2 1 v t u
2 | 2 1 s u v t
t | t u v 1 s 2
u | u v t 2 1 s
v | v t u s 2 1
This shows S_3 is isomorphic to D_3, the dihedral group of degree 3, that
is, the symmetries of an equilateral triangle (this never happens for n
> 3). The lattice of subgroups of S_3 is
S_3
/ / | \
<t> <u> <v> <s>
\ \ | /
{1}
The first three proper subgroups have order two, while <s> has order
three and is the only normal one.The center of S_3 is trivial (in fact Z(S_n) is trivial for all n.) The automorphism group of S_3 is isomorphic to S_3. |
Order 8 (5 groups: 3 abelian, 2 nonabelian)
| C_8
| C_4 x C_2
| C_2 x C_2 x C_2
| D_4, the dihedral group of degree 4, or octic group. It has a
presentation
| <s, t; s^4 = t^2 = e; ts = s^3 t>In terms of these generators (s corresponds to rotation by pi/2 and t to a reflection about an axis through a vertex), the eight elements are 1,s,s^2,s^3,t,ts,ts^2 and ts^3. Using the notation 2 = s^2, 3 = s^3, t2 = ts^2 and t3 = ts^3, the Cayley table is | 1 s 2 3 t ts t2 t3
--+------------------------
1 | 1 s 2 3 t ts t2 t3
s | s 2 3 1 t3 t ts t2
2 | 2 3 1 s t2 t3 t ts
3 | 3 1 s 2 ts t2 t3 t
t | t ts t2 t3 1 s 2 3
ts |ts t2 t3 t 3 1 s 2
t2 |t2 t3 t ts 2 3 1 s
t3 |t3 t ts t2 s 2 3 1
Its subgroup lattice is
D_4
/ | \
{1,s^2,t,ts^2} <s> {1,s^2,st,ts}
/ | \ | / | \
<ts^2> <t> <s^2> <st> <ts>
\ \ | / /
{1}
Of these, the proper normal subgroups are the three of order four
and <s^2> of order two.The center of D_4 is {1,s^2}, which is also its derived group. The automorphism group of D_4 is isomorphic to D_4.
Q, the quaternion group. It has a presentation
| <s, t; s^4 = 1, s^2 = t^2, sts = t>Q can be realized as consisting of the eight quaternions 1, -1, i, -i, j, -j, k, -k, where i is the imaginary square root of -1, and j and k also obey j^2 = k^2 = -1. These quaternions multiply according to clockwise movement around the figure i
/ \
k ---- j
For example, ij = k and ji = -k (negative because anticlockwise).A matrix representation is given by s and t in the above presentation corresponding to these two 2x2 matrices over the complex numbers: s = [i 0] t = [0 i]
[0 -i] [i 0]
The subgroup lattice of Q is
Q
/ | \
<s> <st> <t>
\ | /
<s^2>
|
{1}
All of these subgroups are normal in Q.The center of Q is {1,s^2}, which is also its derived group. The automorphism group of Q is isomorphic to S_4. |
Order 9 (2 groups: 2 abelian, 0 nonabelian)
| C_9
| C_3 x C_3 | |
Order 10 (2 groups: 1 abelian, 1 nonabelian)
| C_10
| D_5 | |
Order 12 (5 groups: 2 abelian, 3 nonabelian)
| C_12
| C_6 x C_2
| A_4, the alternating group of degree 4, consisting of the even
permutations in S_4. The subgroup lattice of A_4 is
| A_4
/ \ \ \ \
<(12)(34),(13)(24)> <(123)> <(124)> <(134)> <(234)>
/ | \ | / / /
<(12)(34)> <(13)(24)> <(14)(23)> | / / /
\ \ \ / / / /
{1}
The only proper normal subgroup is <(12)(34),(13)(24)>.
D_6, isomorphic to S_3 x C_2 = D_3 x C_2
| T which has the presentation
| <s, t; s^6 = 1, s^3 = t^2, sts = t>T is the semidirect product of C_3 by C_4 by the map g : C_4 -> Aut(C_3) given by g(k) = a^k, where a is the automorphism a(x) = -x. Another presentation for T is <x,y; x^4 = y^3 = 1, yxy = x>In terms of these generators, using AB for x^A y^B, the Cayley table for T is | 00 10 20 30 01 02 11 21 31 12 22 32
------+-----------------------------------------------
1 = 00| 00 10 20 30 01 02 11 21 31 12 22 32
x = 10| 10 20 30 00 11 12 21 31 01 22 32 02
x^2 = 20| 20 30 00 10 21 22 31 01 11 32 02 12
x^3 = 30| 30 00 10 20 31 32 01 11 21 02 12 22
y = 01| 01 12 21 32 02 00 10 22 30 11 20 31
y^2 = 02| 02 11 22 31 00 01 12 20 32 10 21 30
xy = 11| 11 22 31 02 12 10 20 32 00 21 30 01
x^2y = 21| 21 32 01 12 22 20 30 02 10 31 00 11
x^3y = 31| 31 02 11 22 32 30 00 12 20 01 10 21
xy^2 = 12| 12 21 32 01 10 11 22 30 02 20 31 00
x^2y^2 = 22| 22 31 02 11 20 21 32 00 12 30 01 10
x^3y^2 = 32| 32 01 12 21 30 31 02 10 22 00 11 20
A 2x2 matrix representation of this group over the complex numbers is
given by
[0 i] [w 0 ]
x <--> [i 0] y <--> [0 w^2]
where i is a square root of -1 and w is nonreal cube root of 1, for
example w = e^{2\pi i/3}. |
Order 14 (2 groups: 1 abelian, 1 nonabelian)
| C_14
| D_7 | |
Order 15 (1 group: 1 abelian, 0 nonabelian)
Order 16 (14 groups: 5 abelian, 9 nonabelian)
| C_16
| C_8 x C_2
| C_4 x C_4
| C_4 x C_2 x C_2
| C_2 x C_2 x C_2 x C_2
| D_8
| D_4 x C_2
| Q x C_2, where Q is the quaternion group
| The quasihedral (or semihedral) group of order 16, with presentation
| <s,t; s^8 = t^2 = 1, st = ts^3> The modular group of order 16, with presentation
| <s,t; s^8 = t^2 = 1, st = ts^5>The elements are s^k t^m, k = 0,1,...,7, m = 0,1. The center is {1,s^2,s^4,s^6}. Its subgroup lattice is G
/ | \
<s^2,t> <s> <st>
/ | \ | /
<s^4,t> <s^2t> <s^2>
/ | \ | /
<t> <s^4t> <s^4>
\ | /
{1}
This is the same subgroup lattice structure as for the lattice of
subgroups of C_8 x C_2, although the groups are of course nonisomorphic.The automorphism group is isomorphic to D_4 x C_2 Reference: Weinstein, Examples of Groups, pp. 120-123. The group with presentation
| < s,t; s^4 = t^4 = 1, st = ts^3 >The elements are s^i t^j for i,j = 0,1,2,3. The center of G is {1,s^2,t^2,s^2t^2}. Reference: Weinstein, pp. 124--128. The group with presentation
| <a,b,c; a^4 = b^2 = c^2 = 1, cbca^2b = 1, bab = a, cac = a> The group G_{4,4} with presentation
| <s,t; s^4 = t^4 = 1, stst = 1, ts^3 = st^3 > The generalized quaternion group of order 16 with presentation
| <s,t; s^8 = 1, s^4 = t^2, sts = t > |
Order 18 (5 groups: 2 abelian, 3 nonabelian)
| C_18
| C_6 x C_3
| D_9
| S_3 x C_3
| The semidirect product of C_3 x C_3 with C_2 which has the presentation
| <x,y,z; x^2 = y^3 = z^3 = 1, yz = zy, yxy = x, zxz = x> |
Order 20 (5 groups: 2 abelian, 3 nonabelian)
| C_20
| C_10 x C_2
| D_10
| The semidirect product of C_5 by C_4 which has the presentation
| <s,t; s^4 = t^5 = 1, tst = s> The Frobenius group of order 20, with presentation
| <s,t; s^4 = t^5 = 1, ts = st^2>
This is the Galois group of x^5 -2 over the rationals, and can be
represented as the subgroup of S_5 generated by (2 3 5 4) and (1 2 3 4 5). |
Order 21 (2 groups: 1 abelian, 1 nonabelian)
| C_21
| <a,b; a^3 = b^7 = 1, ba = ab^2> This is the Frobenius group of
order 21, which can be represented as the subgroup of S_7 generated by (2
3 5)(4 7 6) and (1 2 3 4 5 6 7), and is the Galois group of x^7 - 14x^5 +
56x^3 -56x + 22 over the rationals (ref: Dummit & Foote, p.557). | |
Order 22 (2 groups: 1 abelian, 1 nonabelian)
| C_22
| D_11 | |
Order 24 (15 groups: 3 abelian, 12 nonabelian)
| C_24
| C_2 x C_12
| C_2 x C_2 x C_6
| S_4
| S_3 x C_4
| S_3 x C_2 x C_2
| D_4 x C_3
| Q x C_3
| A_4 x C_2
| T x C_2
| Five more nonabelian groups of order 24 | Reference: Burnside, pp. 157--161. |
Order 25 (2 groups: 2 abelian, 0 nonabelian)
| C_25
| C_5 x C_5 | |
Order 26 (2 groups: 1 abelian, 1 nonabelian)
| C_26
| D_13 | |
Order 27 (5 groups: 3 abelian, 2 nonabelian)
| C_27
| C_9 x C_3
| C_3 x C_3 x C_3
| The group with presentation
| <s,t; s^9 = t^3 = 1, st = ts^4 > The group with presentation
| <x,y,z; x^3 = y^3 = z^3 = 1, yz = zyx, xy = yx, xz = zx>Reference: Burnside, p. 145. |
Order 28 (4 groups: 2 abelian, 2 nonabelian)
| C_28
| C_2 x C_14
| D_14
| D_7 x C_2 | |
Order 30 (4 groups: 1 abelian, 3 nonabelian)
| C_30
| D_15
| D_5 x C_3
| D_3 x C_5 | Reference: Dummit & Foote, pp. 183-184. |