MA1057
Exercise Set 6
Group actions
(1) In each of the following cases, check if the given map defines an action of a group G on the plane
R2 and sketch the orbits for the following points: (x, y) ∈ {(0, 0), (1, 0), (0, 1), (1, 1), (1, 2), (1, 4)}.
The notation t·(x, y) = (xt , yt ) may come in handy throughout. Then Orb (x, y) = {(xt , yt ); t ∈ G}.
a) Translation. (R, +) × R2 → R2 given by t · (x, y) := (x + t, y + 2t), for all t ∈ R, (x, y) ∈ R2 .
b) Rotation. (R, +) × R2 → R2 given by
x
a cos t −b sin t
x
t·
:=
, for all t ∈ R, (x, y) ∈ R2 .
y
b sin t a cos t
y
(2)
(3)
(4)
(5)
For which values of a, b ∈ R is this an action?
d) (R∗ , ·) × R2 → R2 given by t · (x, y) := (tx, t2 y), for all t ∈ R, (x, y) ∈ R2 .
e) (R∗ , ·) × R2 → R2 given by t · (x, y) := (tx, t−1 y), for all t ∈ R, (x, y) ∈ R2 .
f) The multiplicative action of GL2 (R) on R2 .
g) The multiplicative action of O2 (R) on R2 .
For each of the actions G × X → X listed in the previous question, describe X/G as a set: Find a
set of points Y in plane which intersects each of the orbits exactly once.
∗ Let GL (Z) denote the group of multiplicative matrices A with entries in Z which admit an
2
inverse A−1 with entries in Z. (This insures that GL2 (Z) is a group).
Consider the multiplicative
1
a
2
action of GL2 (Z) on Z . Prove that the orbit of the vector
consists only of vectors
0
b
where gcd(a, b) = 1. Can you describe the other orbits of this action?
Consider the following actions G × X → X.
a) (Z, +) × R → R given by (n, x) → x + n, for all n ∈ Z, x ∈ R. Prove that R/Z is bijective to
a circle of circumference 1.
b) (5Z, +) × Z → Z given by (5n, x) → x + 5n, for all 5n ∈ 5Z, x ∈ Z. Prove that Z/5Z is
bijective to Z5 .
c) (Z, +) × R2 → R2 given by (n, (x, y)) → (x + n, y), for all n ∈ Z, (x, y) ∈ R2 . Prove that R2 /Z
is bijective to a cylinder of circumference 1.
d) ∗ (Z2 , +) × R2 → R2 given by ((m, n), (x, y)) → (x + m, y + n), for all (m, n) ∈ Z2 , (x, y) ∈ R2 .
Prove that R2 /Z2 is bijective to a torus (doughnut).
e) (S2 , +) × R2 → R2 given by (σ, (x1 , x2 ) → (xσ−1 (1) , xσ−1 (2) ). Prove that R2 /S 2 is bijective to
a half plane, (which is in its turn bijective to R2 ).
Consider the action of the group G on itself by conjugation:
G×G → G
(g, x) → gxg −1 .
For each x ∈ G, let [x] denote the orbit of x. These orbits are called conjugacy classes. The
centralizer of x is CG (x) := {g ∈ G; gx = xg}. The centre of G is Z(G) = {x ∈ G; gx = xg, ∀g ∈
G}.
a) Prove that CG (x) is the stabilizer of x under the conjugation.
b) Prove that Z(G) G and x ∈ Z(G) ⇐⇒ [x] = {x}.
c) The class equation. Use the equation of orbits to prove that
X |G|
,
|G| = |Z(G)| +
|CG (x)|
[x]
1
2
(6)
(7)
(8)
(9)
(10)
(11)
(12)
(13)
(14)
(15)
where the sum is taken after all the orbits [x] with more than one element.
Consider the action of S3 on itself by conjugation: g · x := gxg −1 . Describe the orbits of (12) and
(123) under this action. How many orbits are there in total in S3 ? Find the stabilizer of each of
the elements above.
Consider the action of S4 on itself by conjugation: g · x := gxg −1 . Describe the orbits of the
following elements under this action: (12), (123), (12)(34). How many orbits are there in total in
S4 ? Find the stabilizer of each of the elements above.
Consider the action of A4 on itself by conjugation: g · x := gxg −1 . Describe the orbits of the
following elements under this action: (123) and (132). How many orbits are there in total in A4 ?
Let G be a non-commutative group and p a prime divisor of |G|. Use the class equation and
induction on |G| to solve the following:
a) Cauchy’s theorem. Prove that there exists an element x ∈ G of order p.
b) If |G| = pn prove Z(G) 6= {e}. For any 0 < k ≤ n, prove that there exists H G with |H| = pk .
c) If |G| = p2 , prove that G ∼
= C2p or G ∼
= Cp × Cp .
2
q−1
Here Cq := {e, a, a , ..., a } denotes the cyclic group of order q.
Let G be a group with |G| = 2p.
a) Prove that G has two generators s and r or orders 2 and p, respectively.
b) If G is commutative, prove that G is cyclic.
2
c) If G is not commutative, prove that srs−1 = ri for some i. Prove that ri = r and hence
p|(i2 − 1). Deduce that G is isomorphic to the dihedral group D2p .
Let H be a subgroup of G such that |G/H| = 2. Then prove that H is a normal subgroup of G.
Recall Lagrange’s Theorem and its Corollary about the order of an element in a finite group. Use
this to prove Fermat’s Little Theorem: ap−1 ≡ 1 (mod p) for any p prime, a not a multiple
of p.
Apply Fermat’s little theorems to simplify 72012 in Z5 ;
Z11 .
Find all normal subgroups of S3 . (Hint: if a subgroup contains an element, then it contains the
entire conjugacy class of the element).
[Hint: If a normal subgroup contains a cycle, then it contains all cycles of the same length. This
limits the options for normal subgroups.]
Consider N S4 .
a) If N contains a transposition, prove that N = S4 .
b) If N contains a cycle of length 3, prove that N ⊆ A4 .
c) If N contains a cycle of length 4, prove that N = S4 .
d) Find (with proof) all the normal subgroups of S4 .
[Hint for b) What are all the elements of A4 ? What is (123)(124) in cycle notation (i.e. using
disjoint cycles)? c) Conjugate and multiply away until you get all types of permutations in S4 . ]
∗ A group having no normal subgroup other than {e} and itself is called simple. The aim of this
exercise is to prove An is simple for n ≥ 5. Recall that An is the subgroup of even permutations
and as such, it’s generated by length 3 cycles.
Consider α ∈ An and let length α := |{i ∈ {1, 2, ..., n}; α(i) 6= i}|.
If length α > 4, then α’s cycle notation starts with at least 5 numbers i1 i2 i3 i4 i5 , (meaning that
α := (i1 i2 )(i3 i4 i5 ...)... or α := (i1 i2 i3 )(i4 i5 ...)... or α := (i1 i2 )(i3 i4 )(i5 ...)..., etc.)
a) Let β := (i3 i4 i5 )α(i3 i4 i5 )−1 α−1 . Prove that length β < length α.
b) If N An and N 6= {e}, prove that N contains a length 3 cycle.
c) Let γ be any length 3 cycle. If n ≥ 5, prove there exists a transposition τ such that τ γτ −1 = γ.
Recall that all length 3 cycles can be written in the form σγσ −1 for some σ ∈ Sn . Prove that
the same is true when we restrict to σ ∈ An .
d) If n ≥ 5 and N An , prove that N = {e} or N = An .
3
[Hints: a) Find some k such that β(ik ) = ik . b) Use decreasing induction on the length of cycles
in N . d) Use b), c), and the fact that An is generated by length 3 cycles.]
(16) ∗ Correspondence theorem. Let f : G Ḡ be a surjective (onto) group homomorphism.
Check that the following map is a one-to-one correspondence
{ Subgroups of Ḡ} → { Subgroups of G containing Ker f },
H̄ → f −1 (H̄) := {x ∈ G; f (x) ∈ H̄},
and that it sends normal subgroups to normal subgroups. As applications:
a) Use the group homomorphism sign : Sn → ({−1, 1}, ·) and Ex.3 to find all normal subgroups
of Sn for n ≥ 5.
b) Let {1, 2, 3, 4} denote the vertices of a tetrahedron (triangular piramid). Let
X = {(12)(34), (13)(24), (23)(14)}
be the set of pairs of opposite sides in the tetrahedron. The action of S4 on the vertices of the
tetrahedron induces an action of S4 on X. As |X| = 3, this is equivalent to a map f : S4 → S3 .
Use this map to find the normal subgroups
of S4 .
T
T
[Hint for a) If N Sn then check N An An , hence either N An = {e} or An ⊆ N . Prove
that the first case is only possible if N = {e} and in the second case, use the Correspondence
theorem. ]
(17) ∗ For any group G and any subgroup H, define the normalizer of H in G by
NG (H) = {g ∈ G; gHg −1 ⊆ H}.
Prove the following:
a) NG (H) is a subgroup of G, and H G ⇐⇒ G = NG (H).
b) H is a normal subgroup of NG (H).
T
c) If P ⊆ NG (H) is a subgroup, then HP is a subgroup of G and H P P and H HP and
P
∼ HP .
T
=
(H P )
H
in particular, this is always true if H is a normal subgroup of G.
G
(18) ∗ For any group G and any subgroup H, let H
be the quotient under the right action of H on G.
The natural group action
G
G
→
,
G×
H
H
(g1 , g2 H) → g1 g2 H,
G
induces a group homomorphism f : G → Sym H
.
T
a) Prove that ker f = g∈G gHg −1 , the largest normal subgroup contained in H.
|G|
|H| = p.
|G|
|H| = p is
b) Let
Prove that
|G|
| ker f |
divides p!
c) If
the smallest prime in the prime factorization of |G|, prove that H = ker f and
hence H is a normal subgroup of G.