Factor Group

• Let be a normal subgroup of .

  The factor group (also called quotient group of modulo , denoted is the group formed from the coset of (i.e. the set ) where

  It has an isomorphism defined as

  Both coset multiplication and the isomorphism are independent of .

  See (Fraleigh 14.4) for a proof that being a normal subgroup is necessary and sufficient for coset multiplication to be well defined.

  • In effect, this simplifies the group structure by representing all Cosets with a representative element.
  • (Fraleigh 14.5) The set of cosets forms a group
    • In the factor group , the coset serves as .
  • (Fraleigh 14.1) One natural choice for is the Kernel of the homomorphism

• For any group and

• The larger the normal group , the smaller the factor group

=

• (Fraleigh 14.9) Let . Then where

  is a homomorphism with kernel .

• (Fraleigh 15.8) Let . Then, is a normal subgroup of . Also, in a natural way. Similarly, in a natural way. ¹

  • For take the homomorphism .
  • For take the homomorphism

• (Fraleigh 15.9) The factor group of a cyclic group is cyclic.

Links §

• A First Course in Abstract Algebra 7th Edition by Fraleigh

Footnotes §

1. See Direct Product of Groups ↩