Discrete Mathematical Structures by Pearson

7-34 Discrete Mathematical Structures

Hence, we find that except the coset H, which is both a left coset and a right coset, a left coset

is not the right coset.

Definition 7.6 The number of left (right) cosets of H in G is called the index of H in G and is

denoted by (G:H).

For example 2, (S

3

:H) = 3

7.9.2 Normal Subgroups

A subgroup H of a group G is called a normal subgroup if aH = Ha for every a ∈ G, that is, if

each left coset is also the right coset.

If G is abelian, then all its subgroups are normal subgroups. However, if G is not abelian, yet

it may have a subgroup such that aH = Ha for every a ∈ G.

Note 7.6 Every group G has at least two normal subgroups, namely G itself and the subgroup

consisting of the identity element e alone.

Definition ...