Group Theory

Question 1

Conjugation of an element x

Answer 1

g(x) = gxg^-1

Question 2

For a set G acting on a set X,

the orbit of x in X

Answer 2

Gx = { g(x) | g in G }

Question 3

For a set G acting on a set X,

what is the orbit of x in X equivalent to

Answer 3

Gx == the conjugacy class of x

Question 4

Rotation group of a cube

Answer 4

Equivalent to S/4

(subscript 4)

Question 5

Rotation group of a tetrahedron

Answer 5

Equivalent to A/4

(subscript 4)

Full symmetry group equivalent to S/4

Question 6

Burnside’s lemma for colourings of a polygon

Answer 6

Number of colourings = avg. size of the fixed set

Question 7

For g in a group G, and a set X,

X/g

(subscript g)

Answer 7

X/g = { x in X | g(x) = x }

(subscript g)

Question 8

What is the order of a composition of cycles equivalent to

Answer 8

l.c.m. (cycle lengths)
or
product (cycle lengths), if all are coprime

Question 9

Subgroup

Answer 9

A subgroup of a group (G, x) is a subset H satisfying:
* if h,k is in H, then h x k is in H
* identity element of G is in H
* inverse of h in H is also in H

Question 10

Homomorphism equation

Answer 10

Question 11

Bezout’s identity

Answer 11

For all m,n in Z, there exists a,b in Z s.t.
am + bn = gcd(m,n)

Question 12

Lagrange’s theorem

Answer 12

If H is a subgroup of a finite group G, then the order of H divides the order of G

Question 13

Hamilton’s equations

Quaternions

Answer 13

i^2 = j^2 = k^2 = -1
i . j . k = -1