Does Good Corporate Governance Include Employee Representation ...
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- expression écrite
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- management board
- bank representatives
- employee representation
- supervisory board
- allocation of employees among affiliated firms
- firms
- corporate governance
- firm
- employees
MA441: Algebraic Structures I
Lecture 25
8 December 2003
1Review from Lecture 24:
Internal Direct Products
Notation: for subgroups H,K <G,
HK ={hk|h∈H,k ∈K}.
Definition:
We say that G is the internal direct product
of H and K and write G=H K
if H,KCG and
G=HK and H ∩K ={e}.
2Definition:
Let H ,H ,...,H be a finite collection of nor-n1 2
mal subgroups of G. We say that G is the
internal direct product of H ,H ,...,H andn1 2
write
G=H H Hn1 2
if the following two conditions hold:
1. G=H H H ={h h h |h ∈H },n n1 2 1 2 i i
2. (H H H )∩H ={e}(i=1,...,n 1).1 2 i i+1
3Theorem 9.6
If a group G is the internal direct product of
a finite number of subgroups H ,H ,...,H ,n1 2
then G is isomorphic to the external direct
product of H ,H ,...,H .n1 2
4Example: (p. 185) Let m=n n n , where1 2 k
the n are relatively prime to each other. Pre-i
viously we saw that
U(m)U(n )U(n )U(n ).1 2 k
This external direct product is also an internal
direct product:
U(m)U (m)U (m)U (m).m/n m/n m/n1 2 k
For example,
U(105) U(7)U(15)
= U (105)U (105)15 7
= {1,16,31,46,61,76}
{1,8,22,29,43,64,71,92}
5Definition:
A homomorphism from a group G to a1
group G is a mapping from G to G that2 1 2
preserves the group operation; that is, for all
a,b∈G,
(ab)=(a)(b).
6Definition:
The kernel of a homomorphism : G → G1 2
is the set {x∈G|(x)=e}.
We denote the kernel of by Ker.
Examples:
ThekernelofthedeterminantmapfromGL(2,R)
toR isthesubgroupofmatriceswithdetermi-
nant 1 is SL(2,R). (This is called the special
linear group).
The kernel of the derivative map on polynomi-
als is the subgroup of constant polynomials.
7Theorem 10.1
Let :G →G be a homomorphism.1 2
Let g be in G . Then1
1. sends the identity of G to the identity1
of G .2
A homomorphism preserves identity.
n n2. (g )=(g) (∀n∈Z)
A homomorphism preserves powers.
83. If |g| is finite, then |(g)| divides |g|.
Thehomomorphicimageofanelement
has an order that divides the order of
that element.
4. Ker<G.
The kernel of a homomorphism is a
subgroup.
5. If (g )=g , then1 2
1 (g )={x∈G |(x)=g }=g Ker.2 1 2 1
The homomorphic preimage of an ele-
ment is a coset of the kernel.
9Theorem 10.2:
Let : G → G be a homomorphism and let1 2
H <G . We have the following properties:1
1. (H)={(h)|h∈H} is a subgroup of G .2
The homomorphic image of a subgroup
is a subgroup, or
A homomorphism preserves the prop-
erty of being a subgroup.
2. If H is cyclic, then (H) is cyclic.
The homomorphic image of a cyclic
group is cyclic, or
A homomorphism preserves the prop-
erty of being cyclic.
10
