the fate of the memory trace

the fate of the memory trace

English
17 Pages
Read
Download
Downloading requires you to have access to the YouScribe library
Learn all about the services we offer

Description

  • cours - matière potentielle : students
  • mémoire
  • mémoire - matière potentielle : tracers
  • mémoire - matière potentielle : for decades
  • cours - matière potentielle : diary
  • mémoire - matière potentielle : processes
  • cours magistral - matière potentielle : on state of the art research
  • mémoire - matière potentielle : trace
  • cours magistral - matière potentielle : by experts
  • cours magistral
  • mémoire - matière potentielle : systems
  • mémoire - matière potentielle : trace 50 ece faculty of the third week
  • mémoire - matière potentielle : theories
  • mémoire - matière potentielle : traces
  • cours - matière potentielle : the ece
  • mémoire - matière potentielle : system
  • mémoire - matière potentielle : research
the fate of the memory trace european campus of excellence summer school in neuroscience at rub
  • successful history of the european campus of excellence
  • fate of the memory trace
  • initiatives
  • summer school
  • step by step
  • germany
  • university
  • research
  • students

Subjects

Informations

Published by
Reads 18
Language English
Report a problem

MA441: Algebraic Structures I
Lecture 16
29 October 2003
1Review from Lecture 15:
Theorem 6.1: Cayley’s Theorem
Every group is isomorphic to a group of per-
mutations.
Example:
(RD1)= (132)(23) =((132))((23))( )
((132))((23))=(123)(456)(14)(26)(35)
(123)(456)(14)(26)(35)=(16)(25)(34)
(16)(25)(34)=((12))=(D3).
2Theorem 6.2: Properties of Isomorphisms
Acting on Elements
Suppose that : G → G is an isomorphism.1 2
Then the following properties hold.
1. sends the identity of G to the identity1
of G .2
2. For every integer n and for every group
n nelement a in G , (a )=((a)) .1
3. For any elements a,b ∈ G , a and b com-1
mute iff (a) and (b) commute.
4. The order of a, |a| equals |(a)| for all
a∈G (isomorphisms preserve orders).1
35. For a fixed integer k and a fixed group ele-
kment b in G , the equation x =b has the1
same number of solutions in G as does1
kthe equation x =(b) in G .2
Proof:
Part 5:
Apply the isomorphism to the equation
k k kx =b to get (x )=(x) =(b).
Let’s rename the variable x to y in the second
kequation and write y =(b).
For every solution x ∈ G to the first equa-1
tion, we get a solution y ∈ G to the second2
equation. Because is one-to-one, there are
at least as many y as x.kSuppose y ∈ G is a solution to y = (b).2
Since is onto, there is an x ∈ G such that1
(x)=y.
k k kNow y = (x) = (x ) = (b). Since is
kone-to-one, we know x =b.
Therefore we have at least as many x as y, and
the number of solutions of the two equations
are equal.
(Non)example: C is not isomorphic to R
4because the equation x = 1 has a different
number of solutions in each group.
4Theorem 6.3: Properties of Isomorphisms
Acting on Groups
Suppose that : G → G is an isomorphism.1 2
Then the following properties hold.
1. G is Abelian iff G is Abelian.1 2
2. G is cyclic iff G is cyclic.1 2
13. is an isomorphism from G to G .2 1
4. If K G is a subgroup, then (K) =1
{(k)|k ∈K} is a subgroup of G .2
5Proof:
Part 1: follows from part 3 of Theorem 6.2,
which shows that isomorphisms preserve com-
mutativity.
Part 2: follows from part 4 of Theorem 6.2,
which shows that isomorphisms preserve order
and by noting that if G =hai, then1
G =h(a)i.2
6Part 3: Since is one-to-one and onto, for
every y ∈ G , there is a unique x ∈ G such2 1
1that (x)=y. Define (y) to be this x.
1Clearly, is one-to-one and onto, since is.
1In fact, is the identity map on G , and2
1 is the identity map on G .1
We need to show the homomorphism property
1for :
1 1 1 (ab)= (a) (b).
71Let (x)=a (so (a)=x) and
1let (y)=b (so (b)=y).
Then substituting for a and b,
1 1 (ab) = ((x)(y))
1= ((xy))
= xy
1 1= (a) (b).
1Therefore :G →G is an isomorphism.2 1
8Definition:
An isomorphism from a group G onto itself is
called an automorphism of G. The set of
automorphisms is denoted Aut(G).
Example 9:
Complex conjugation is an automorphism of C
under addition and C under multiplication.
Example 10:
2In R , (a,b) = (b,a) is an automorphism of
2R under componentwise addition.
9