# The Best Introduction To Representation Theory

is Cayley's theorem. I don't know why textbooks don't emphasize th
is. Everyone should learn this while studying group theory for the
first time.


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Thm (Cayley's theorem). Every group is isomorphic to a subgroup of
a symmetric group.
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pf) Set m: G -> S(G), m(x) = m_x where m_x: G -> G, m_x(y) = x*y.
Then m is an injective homomorphism. In other words, G is isomorph
ic to a subgroup of S(G).


Then for finite cases, you already know that S(n) is isomorphic to
the matrix group of n*n permutation matrices.


Ex) S(3) is isomorphic to the group of 3*3 permutation matrices,

|1 0 0| |0 0 1| |0 1 0| |0 1 0| |0 0 1| |1 0 0|
|0 1 0| |1 0 0| |0 0 1| |1 0 0| |0 1 0| |0 0 1|
|0 0 1| |0 1 0| |1 0 0| |0 0 1| |1 0 0| |0 1 0|
.


So you already know that every finite group is isomorphic to a mat
rix group. In other words, you can represent every group element a
s a matrix.

This also means that you can regard every group element as a trans
formation acting on a space. Since you can regard a matrix as a co
ordinate of a linear map.