<a, b; a2 = b2 = (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]A permutation representation is the following four elements of S4:
[0 1], [0 -1], [ 0 1], [ 0 -1]
(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}
<s,t; s2 = t2 = 1, sts = tst>
Another presentation (with s <-> (1, 2, 3), t <-> (1, 2)) is
<s,t; s3 = t2 = 1, ts = s2 t>
In terms of this second presentation, with 2 = s2, u = ts
and v = ts2, 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 S3 is isomorphic to D3, 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 S3 is
S3
/ / | \
<t> <u> <v> <s>
\ \ | /
{1}
The first three proper subgroups have order two, while <s> has
order three and is the only normal one. <s, t; s4 = t2 = e; ts = s3 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,s2,s3,t,ts,ts2 and ts3. Using the notation 2 = s2, 3 = s3, t2 = ts2 and t3 = ts3, 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
D4
/ | \
{1,s2,t,ts2} <s> {1,s2,st,ts}
/ | \ | / | \
<ts2> <t> <s2> <st> <ts>
\ \ | / /
{1}
Of these, the proper normal subgroups are the three
of order four and <s2> of order two.
<s, t; s4 = 1, s2 = t2, 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 j2 = k2 = -1. These quaternions
multiply according to
clockwise movement around the figure
i
/ \
k ---- j
For example, ij = k and ji = -k (negative because
anticlockwise).
s = [i 0] t = [0 i]
[0 -i] [i 0]
The subgroup lattice of Q is
QAll of these subgroups are normal in Q.
/ | \
<s> <st> <t>
\ | /
<s2>
|
{1}
A4
/ \ \ \ \
<(1,2)(3,4),(1,3)(2,4)> <(1,2,3)> <(1,2,4)> <(1,3,4)> <(2,3,4)>
/ | \ | / / /
<(1,2)(3,4)> <(1,3)(2,4)> <(1,4)(2,3)> | / / /
\ \ \ / / / /
{1}
The only proper normal subgroup is
<(1,2)(3,4),(1,3)(2,4)>.
<s, t; s6 = 1, s3 = t2, sts = t>
T is the semidirect product of C3 by C4
by the map g : C4 -> Aut(C3) given by
g(k) = ak, where a is the automorphism
a(x) = -x.
<x,y; x4 = y3 = 1, yxy = x>
In terms of these generators, using AB for xA yB, 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
x2 = 20| 20 30 00 10 21 22 31 01 11 32 02 12
x3 = 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
y2 = 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
x2y = 21| 21 32 01 12 22 20 30 02 10 31 00 11
x3y = 31| 31 02 11 22 32 30 00 12 20 01 10 21
xy2 = 12| 12 21 32 01 10 11 22 30 02 20 31 00
x2y2 = 22| 22 31 02 11 20 21 32 00 12 30 01 10
x3y2 = 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 w2]
where i is a square root of -1 and w is nonreal cube root of
1, for
example w = e2\pi i/3.
<s,t; s8 = t2 = 1, st = ts3 >
<s,t; s8 = t2 = 1, st = ts5 >
G
/ | \
<s2,t> <s> <st>
/ | \ | /
<s4,t> <s2t> <s2>
/ | \ | /
<t> <s4t> <s4>
\ | /
{1}
This is the same subgroup lattice structure as for the
lattice of subgroups of C8 x C2, although
the groups are of course nonisomorphic. < s,t; s4 = t4 = 1, st = ts3 >SmallGroup id: ???????
<a,b,c; a4 = b2 = c2 = 1, cbca2b = 1, bab = a, cac = a>
<x,y,z; x2 = y3 = z3 = 1, yz = zy, yxy = x, zxz = x>
<s,t; s4 = t5 = 1, tst = s>
<s,t; s4 = t5 = 1, ts = st2>
This is the Galois group of x5 -2 over the rationals, and can be
represented as the subgroup of S5 generated by (2, 3, 5, 4) and
(1, 2, 3, 4, 5).
<s,t; s9 = t3 = 1, st = ts4 >
<x,y,z; x3 = y3 = z3 = 1,
yz = zyx, xy = yx, xz = zx>
Reference: Burnside, p. 145.