Cosets

Proposition (Right Cosets) If is a subgroup of , then the relation defined on by if and only if is an equivalence relation of

    Proof. If then because If then and so implies If and then and and so Thus, is reflexive, symmetric, and transitive as desired.

Definition (Right Cosets) If is a subgroup of , then the set of equivalence classes for the equivalence relation defined by if and only if are called the right cosets of in

    The right coset of to which belongs is denoted Indeed, if the group operation is then cosets are denoted by In particular,  if the operation is addition or multiplication, then we denote cosets by or respectively.

Proposition (Cosets of a Subgroup) If is a subgroup of and then the following are equivalent conditions:

    (i)
    
    (ii) for some
    
    (iii)
    
    (iv)
    
Moreover, if is finite, then

    Proof. If then for some and so for some
     If   for some then and so which means that is the right coset of that is
     Suppose or equivalently . If then and so by transitivity. Therefore, Conversely, follows by symmetry.
     Given then is in the right coset of and conversely. Thus, which means   
    Suppose is finite and define the mapping by for each This mapping is well-defined by the operation on Further, is one-to-one because if then and the right cancellation property, implies Also, is onto since consists of elements of the form for some by parts (i)-(iv).

     One right coset of in will be To compute all the right cosets of in a finite group first choose any element in the complement of and compute Then choose any element and compute If we continue until is exhausted we have computed all right cosets of   

Example (Right Cosets) Let and Then,

are the right cosets of It is important to notice that these right cosets partition the group.

Example (Distinguish Between Left and Right Cosets) Let be the group of all permutations on a set with three elements where and and let Then the left cosets of are and and the right cosets of are

Proposition (Lagrange's Theorem) If is a subgroup of a finite group then the order of is a divisor of the order of

    Proof. Given a subgroup we can form the set of all right cosets of say there are of them, and we can choose one element from each coset say Then since all the right cosets partition Since and for any it follow that Thus,  the order of is a divisor of the order of as desired.

Proposition (Consequences of the Lagrange Theorem)

    (i) If is a finite group with then
    
    (ii) Any group of prime order contains no subgroup other than and is cyclic, and is generated by one of its nonidentity elements.
    
    Proof. (i) Since and is a subgroup Therefore, for some integer
    (ii) Since a prime only has 1 and itself as a divisor there are no subgroups of besides and If then and so is cyclic as desired.

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