Week 9

Question 1

A group J is a semi direct product of a subgroup H by a subgroup G is the following conditions are satisfied

Answer 1

Question 2

Relate direct product to semi direct product

Answer 2

Direct product is special case of semi direct product where both subgroups G and H are normal subgroups of J

Question 3

Answer 3

Denoted that H is a normal subgroup of J

Question 4

Answer 4

Denotes that H is a normal subgroup of (above) and the semi direct product can act on the set G x H

(Combination of direct product and normal subgroup notations)

Question 5

If J is the extension of H by G

Answer 5

A group J is a semi direct product of a subgroup H by a subgroup G

Question 6

Answer 6

Question 7

Prove

Answer 7

Question 8

Prove

Answer 8

Question 9

Prove

Answer 9

Question 10

Prove

Answer 10

Question 11

Define Euclidean group

Answer 11

Question 12

All transformations which preserve the Euclidean inner product are formed

Answer 12

Of just the orthogonal transformations and translations of R^N

Question 13

Prove

Answer 13

Question 14

Prove

Answer 14

(Final line)
€ O(N), so that Q is a map constructed from a translation and an orthogonal transformation

Question 15

Euclidean group is

Answer 15

Question 16

Prove

Answer 16

Question 17

Poincaré group is

Answer 17

Question 18

Minkowski metric

Answer 18

Question 19

Lorentz transformation

Answer 19

Question 20

The Poincaré group is isomorphic to?
With homomorphism?
Automorphism?

Answer 20