ABSTRACT ALGEBRAIC STRUCTURES OPERATIONS AND ...

102. Definition: The equivalence relation induced by a subgroup
The equivalence relation established in the preceding theorem is said to be the
left equivalence relation induced by H. The equivalence classes are said to be
the left cosets for H.)
It is obvious to define right equivalence relation and right cosets analogously.
Lets give some interpretation of what is going on here. Imagine that the elements
in G have some property (call it their color). You should think of the
elements h in the subgroup H to be so that their action doesn’t change the
colour, by what we mean that ah has the same colour as a. Then the coset aH
consists of the elements with the same colour. If H consists of all the ”neutral”
elements then each coset represents a specific colour. Here are two examples
X40 and X41
103. Theorem: A Left coset for a subgroup is the image of H by a
bijection.
The left cosets have the form aH and are the result of a left translation of H.
The mapping h ↦→ ah is a bijection of H on aH.
Similar for right cosets.
Proof : Let ∼ denote the left equivalence corresponding to H. This means
per definition that b ∼ a ⇔ b ∈ aH. This shows that [a] = aH. Let La denote
left translation with a that means that La(x) = ax. Then La(H) = aH. Left
translation with an invertible element in a monoid is injective (T85). And in
a group all elements are invertible .
You have maybe met this result in the situation in X42. X43
104. Definition: Index for subgroup
If H is a subgroup in G then we define the index for H to be the number
(possibly ∞) of left cosets and we denote it by [G : H]. Notice that we would
have the same number if we had used the right cosets
X44 X45
105. Theorem: Lagrange’s theorem
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