Group theory

Question 1

group

Answer 1

a group G is a set equipped with an operation (group multiplication law)
. or ֯º or *
º : G xG -> G such that
i) if x,y in G then xºy in G (closure)
ii) (xºy)ºz = xº(yºz) for all x,y,z in G (assosiativity)
iii) G contains the identity element e st xºe = eºx = x for all x in G - e is unique
iv) for all x in G there exists x^-1 in G st xºx^-1 = x^-1ºx = e

Question 2

abelian

Answer 2

a group is abelian if xºy = yºx for all x,y in G

Question 3

subgroup

Answer 3

(H,º) is a subgroup of (G,º) if HcG and (H,º) is a group in its own right

Question 4

isomorphic groups

Answer 4

2 groups G and H are isomorphic if there is a bijection preserving the group operation f(g1º g2) = f(g1)ºf(g2) for all g1,g2 in G, f: G->H
(where a bijection is something that is 1 to 1 and onto)
in particular f(eG) = eH

Question 5

finite group

Answer 5

a finite group G is a group having a finite number of elements

Question 6

order

Answer 6

the order of a group denoted by |G| is the no. elements in G

Question 7

order of an element in a group

Answer 7

let g be in G then the smallest +ve integer n st g^n = e, if such n exists, is the order of g denoted by |g|
the identity (e) has order 1

Question 8

lagrange theorem - subgroups and order

Answer 8

let G be a finite order group and H a subgroup of G then the order of H divides the order of G

Question 9

element order and group order

Answer 9

for every element g in G the order of g divides the order of G

Question 10

isomorphic groups and order of elements

Answer 10

if G and H are isomorphic then the order of any element g in G is equal to the corresponding h = f(g) in H
|g|= |f(g)| is bijection

Question 11

direct product of groups

Answer 11

given 2 groups G and H the direction product GxH is a group defined as follows
- elements of GxH are the ordered pairs (g,h), g in G and h in H
- the operation ºGxH is defined as (g1,h1)º(g2,h2) = (g1ºg2, h1ºh2)

Question 12

order of GxH

Answer 12

the direct product GxH form a group st the order of GxH is m.n where m is the order of G and n is the order of H

Question 13

oder of (g,h) and of g and h

Answer 13

order of (g,h) is the LCM of the order of g and the order of h
|(g,h)| = LCM(|g|,|h|)

Question 14

orthogonal transformations

Answer 14

linear transformations v-> Mv preserving the length and angles between the vectors

Question 15

orthogonal group and transformations

Answer 15

the group of orthogonal transformations in R^n is the orthogonal group