Group Theory Flashcards

1
Q

Conjugation of an element x

A

g(x) = gxg^-1

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2
Q

For a set G acting on a set X,

the orbit of x in X

A

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

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3
Q

For a set G acting on a set X,

what is the orbit of x in X equivalent to

A

Gx == the conjugacy class of x

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4
Q

Rotation group of a cube

A

Equivalent to S/4

(subscript 4)

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5
Q

Rotation group of a tetrahedron

A

Equivalent to A/4

(subscript 4)

Full symmetry group equivalent to S/4

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6
Q

Burnside’s lemma for colourings of a polygon

A

Number of colourings = avg. size of the fixed set

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7
Q

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

X/g

(subscript g)

A

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

(subscript g)

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8
Q

What is the order of a composition of cycles equivalent to

A

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

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9
Q

Subgroup

A

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

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10
Q

Homomorphism equation

A
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11
Q

Bezout’s identity

A

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

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12
Q

Lagrange’s theorem

A

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

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13
Q

Hamilton’s equations

Quaternions

A

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

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