Group Theory Flashcards
Conjugation of an element x
g(x) = gxg^-1
For a set G acting on a set X,
the orbit of x in X
Gx = { g(x) | g in G }
For a set G acting on a set X,
what is the orbit of x in X equivalent to
Gx ==
the conjugacy class of x
Rotation group of a cube
Equivalent to S/4
(subscript 4)
Rotation group of a tetrahedron
Equivalent to A/4
(subscript 4)
Full symmetry group equivalent to S/4
Burnside’s lemma for colourings of a polygon
Number of colourings = avg. size of the fixed set
For g in a group G, and a set X,
X/g
(subscript g)
X/g = { x in X | g(x) = x }
(subscript g)
What is the order of a composition of cycles equivalent to
l.c.m. (cycle lengths)
orproduct (cycle lengths)
, if all are coprime
Subgroup
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
Homomorphism equation
Bezout’s identity
For all m,n
in Z
, there exists a,b
in Z
s.t.am + bn = gcd(m,n)
Lagrange’s theorem
If H
is a subgroup of a finite group G
, then the order of H
divides the order of G
Hamilton’s equations
Quaternions
i^2 = j^2 = k^2 = -1
i . j . k = -1