Mathematics

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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