Subgroups and cosets

A subset H of G, H c G, that is itself a group with the same law of binary composition, is a subgroup of G. Any subset of G that satisfies closure will be a subgroup of G, since the other group properties are then automatically fulfilled. The region of the multiplication table of S(3) in Table 1.3 in a box shows that the subset P0 Pi P2 is closed, so that this set is a 

Given a group G with subgroup H c G, then gr H, where gr C G but gr H unless gr is g = E, is called a left coset of H. Similarly, H gr is a right coset of H. The g,., gr C G but gr H, except for g = E, are called coset representatives. It follows from the uniqueness of the product of two group elements (eq. (1.1.2)) that the elements of gr H are distinct from those of gs H when, v / r, and therefore that 

If H gr = gr H, so that right and left cosets are equal for all r, then 

Therefore, H is transformed into itself by all the elements of G H is then said to be an invariant (or normal) subgroup of G. 

Exercise 1.3-1 Prove that any subgroup of index 2 is an invariant subgroup. 

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