Mathematics

1. (10 pts) Show that

a) hsi E

s; r2

E D4 but hsi 5 D4:

b) hsi ‘ Z2 and

s; r2

‘ Z2 Z2 (V4 ‘ Z2 Z2 is called the Kleinís group):

2. (10 pts) Let H be a subgroup of a group G. DeÖne NG (H) ñthe normalizer of H in G as follows

NG (H) =

g 2 G j 8h 2 H; ghg1

2 H

a) Show that NG (H) is a subgroup of G:

b) Let H = f1; sg be a subgroup of G = D6: Find NG (H):

c) Prove that if H E N G then N NG(H): Deduce that NG(H) is the largest subgroup of G in

which H is normal.

3. (10pts) Let H G and K G be Önite subgroups of G:

a) Show that jHKj =

jHjjKj

jH\Kj

; where HK = fhk 2 G j h 2 H and k 2 Kg :

b) Show that HK needs not to be a subgroup of G. Hint: Let G = S3; H = h(1 2)i and K = h(2 3)i:

Show that HK is not a subgroup of G:

c) Show that if H G and K G and H NG (K); then HK is a subgroup of G:

4. (10 pts) Solve the following problems.

a) Show that if H G and jG : Hj = 2 then H E G:

b) Show that Q is not a cyclic group.

5. (10 pts) Show that groups Zn, Dn; Q8 and S4 are solvable. Find the upper central series for D4;

Q8 and S4:

Extra Credit Problems

1. (20 pts) For a group G we denote by Aut (G) the set of all : G ! G such that is an isomorphism

of G, that is

Aut (G) = f j : G ! G is an isomorphismg :

We call Aut (G) the set of all automorphisms of group G:

a) DeÖne a binary operation : Aut (G) Aut (G) ! Aut (G). For every 1; 2 2 Aut (G) let

1 2 : G ! G be deÖned as follows

(1 2) (g) = 1 (2 (g)); for every g 2 G:

Show that (Aut (G); ) is a group.

1

b) Show that Aut (D4) is isomorphic to D4; i.e. Aut (D4) ‘ D4.

2. (30 pts) Let O (3) =

A 2 Mat (3; R) j AT A = I3

the group of orthogonal transformations of R

3

.

Consider

SO (3) = fA 2 O (3) j det A = 1g :

a) Show that SO (3) O (3):

b) Show that elements of SO (3) correspond to rotations of R

3

: Namely, show that, for any A 2 SO (3);

there is a plane through the origin of R

3

such that:

i) The restriction of A to ? =

x 2 R

3

j 8v 2 ; x v = 0

is the identity, that is the linear transformation

Aj? : ? ! R

3

; given by

Aj? (x) = A (x) = x;

that is:

Aj? = I3;

ii) The restriction of A to is a rotation of :

c. Show that SO (3) has a subgroup isomorphic to O (2):

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