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In the following discussion, we will use a binary operation on the subsets of G: if two subsets S and T of G are given, we define their product as:
This operation is associative and has identity element {e}, where e is the identity element of G. Thus, the set of all subsets of G forms a monoid under this operation.
A subgroup N of a group G is normal if and only if the coset equality aN = Na holds for all a in G. In terms of the binary operation on subsets defined above, a normal subgroup of G is a subgroup that commutes with every subset of G.
We define the set G/N to be the set of all left cosets of N in G, i.e.
The group operation on G/N is the product of subsets defined above. In other words, for each aN and bN in G/N, the product of aN and bN is (aN)(bN). For this operation to be closed, we must show that (aN)(bN) really is a left coset:
Note that we have already used the normality of N in this equation. Also note that because of the normality of N, we could have chosen to define G/N as the set of right cosets of N in G. Also note that because the operation is derived from the product of subsets of G, the operation is well-defined (does not depend on the particular choice of representatives), associative and has identity element N.
The inverse of an element aN of G/N is a−1N. This completes the proof that G/N is a group.
Consider the group of integers Z (under addition) and the subgroup 2Z consisting of all even integers. This is a normal subgroup, because Z is abelian. There are only two cosets, the set of even integers and the set of odd integers, and Z/2Z is the cyclic group with two elements.
As another abelian example, consider the group of real numbers R (again under addition) and the subgroup Z of integers. The cosets of Z in R are all sets of the form a + Z, with 0 ≤ a < 1 a real number. Adding such cosets is done by adding the corresponding real numbers, and subtracting 1 if the result is greater than or equal to 1. The factor group R/Z is isomorphic to S1, the group of complex numbers of absolute value 1 under multiplication. An isomorphism is given by f(a + Z) = exp(2πia) (see Euler's identityIn mathematics, Euler's identity a special case of Euler's formula, is the following: : The equation appears in Leonhard Euler's Introduction published in Lausanne in 1748. In this equation, e is the base of the natural logarithm, is the imaginary unit (a).
If G is the group of invertible 3×3 real matricesAbstract algebra Algebra Linear algebra In mathematics, a matrix (plural matrices is a rectangular table of numbers or, more generally, of elements of a fixed ring. In this article, if unspecified, the entries of a matrix are always real or complex number, and N is the subgroup of 3×3 real matrices with determinantIn linear algebra, the determinant is a function that associates a scalar det A to every square matrix A''. The fundamental geometric meaning of the determinant is as the scale factor for volume when A is regarded as a linear transformation. Determinants 1, then N is normal in G (since it is the kernel of the determinant homomorphismGiven two groups G ) and H ·), a group homomorphism from G ) to H ·) is a function h : G H such that for all u and v in G it holds that : h ''u v h ''u · h ''v From this property, one can deduce that h maps the identity element e of G to the identity elem), and G/N is isomorphic to the multiplicative group of non-zero real numbers.
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