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THE PRODUCT OF SUBSETS OF A GROUP

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:

:ST = \{ st : s \isin S { m~and~} t \isin T \}.

This operation is Associative and has as Identity Element the Singleton {''e''}, where ''e'' is the identity element of ''G''. Thus, the set of all subsets of ''G'' forms a Monoid under this operation.

In terms of this operation we can first explain what a quotient group is, and then explain what a normal subgroup is:

A quotient group of a group


It is fully determined by the subset containing ''e''. A Normal Subgroup of ''G'' is the set containing ''e'' in any such partition. The subsets in the partition are the Coset s of this normal subgroup.

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''.


DEFINITION

We define the set ''G''/''N'' to be the set of all left cosets of ''N'' in ''G'', i.e.,

:G/N = \{ aN : a \isin G \}.

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:

:(''aN'')(''bN'') = ''a''(''Nb'')''N'' = ''a''(''bN'')''N'' = (''ab'')''NN'' = (''ab'')''N''.

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''−1''N''. This completes the proof that ''G''/''N'' is a group.


EXAMPLES


  • Consider the group of Integer s 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; therefore, the quotient group Z/2Z is the cyclic group with two elements. This quotient group is isomorphic with the set { 0, 1 } with addition modulo 2; informally, it is sometimes said that Z/2Z ''equals'' the set { 0, 1 } with addition modulo 2.



  • Consider the multiplicative abelian group ''G'' of Complex twelfth Roots Of Unity , which are points on the Unit Circle , shown on the picture on the right as colored balls with the number at each point giving its complex argument. Consider its subgroup ''N'' made of the fourth roots of unity, shown as red balls. This normal subgroup splits the group into three cosets, shown in red, green and blue. One can check that the cosets form a group of three elements (the product of a red element with a blue element is red, the inverse of a blue element is green, etc.). Thus, the quotient group ''G''/''N'' is the group of three colors, which turns out to be the cyclic group with three elements.


  • Consider the group of Real Number s R 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 quotient group R/'''Z''' is isomorphic to the Circle Group S1, the group of Complex Number s of Absolute Value 1 under multiplication, or correspondingly, the group of Rotation s in 2D about the origin, i.e., the special Orthogonal Group SO(2). An isomorphism is given by ''f''(''a'' + '''Z''') = exp(2π''ia'') (see Euler's Identity ).


  • If ''G'' is the group of invertible 3×3 real Matrices , and ''N'' is the subgroup of 3×3 real matrices with Determinant 1, then ''N'' is normal in ''G'' (since it is the Kernel of the determinant Homomorphism ). The cosets of ''N'' are the sets of matrices with a given determinant, and hence ''G''/''N'' is isomorphic to the multiplicative group of non-zero real numbers.


  • Consider the abelian group Z4 = Z/4Z (that is, the set { 0, 1, 2, 3 } with addition Modulo 4), and its subgroup { 0, 2 }. The quotient group Z4 / { 0, 2 } is { { 0, 2 }, { 1, 3 } }. This is a group with identity element { 0, 2 }, and group operations such as

    • ::{ 0, 2 } + { 1, 3 } = { 1, 3 }

:Both the subgroup { 0, 2 } and the quotient group { { 0, 2 }, { 1, 3 } }, with their group operations induced by Cyclic Group Z4, are isomorphic with Z2.


PROPERTIES

Trivially, ''G / G'' is Isomorphic to the Trivial Group (the group with one element), and ''G /'' {e} is isomorphic to ''G''.

The of ''N'' in ''G''. If ''G'' is finite, the index is also equal to the order of ''G'' divided by the order of ''N''. Note that ''G / N'' may be finite, although both ''G'' and ''N'' are infinite (e.g. Z ''/'' 2Z).

There is a "natural" is ''N''.

There is a bijective correspondence between the subgroups of ''G'' that contain ''N'' and the subgroups of ''G / N''; if ''H'' is a subgroup of ''G'' containing ''N'', then the corresponding subgroup of ''G / N'' is π(''H''). This correspondence holds for normal subgroups of ''G'' and ''G / N'' as well, and is formalized in the Lattice Theorem .

Several important properties of quotient groups are recorded in the Fundamental Theorem On Homomorphisms and the Isomorphism Theorem s.

If ''G'' is Abelian , Nilpotent or Solvable , then so is ''G / N''.

If ''G'' is Cyclic or Finitely Generated , then so is ''G / N''.

If ''H'' is a subgroup in a finite group ''G'', and the order of ''H'' is one half of the order of ''G'', then ''H'' is guaranteed to be a normal subgroup, so ''G / H'' exists and is isomorphic to ''C''2. This result can also be stated as "any subgroup of index 2 is normal", and in this form it applies also to infinite groups.

Every group is isomorphic to a quotient of a Free Group .

Sometimes, but not necessarily, a group ''G'' can be reconstructed from ''G / N'' and ''N'', as a Direct Product or Semidirect Product . An example where it is ''not'' possible is as follows. Z4 / { 0, 2 } is isomorphic to Z2, and { 0, 2 } also, but the only semidirect product is the direct product, because Z2 has only the trivial Automorphism . Therefore Z4, which is different from Z2 × Z2, cannot be reconstructed.


SEE ALSO