Group Homomorphism

• Let and be binary algebraic structures. A mapping is a homomorphism if

  • The homomorphism property says that the product of the maps is equal to the map of the products.

• (Fraleigh e13.49) The composition of homomorphisms is a homomorphism.

• (Fraleigh 13.12) Let be a homomorphism.

  • The identity is preserved. If is the identity of , then is the identity of

  • The inverse is preserved.

  • Subgroups are preserved.

  • Subgroup inverses are preserved

  • (Fraleigh 15.16) Normal groups are preserved.

• Let be a homomorphism. The kernel of , denoted is defined as the subgroup

  Where is the identity of .

  • We may think of this as the set of points which the homomorphism maps to a singularity

• (Fraleigh 13.15) If and , then we have the cosets of as

  It follows that the partition of into cosets is the same.

• (Fraleigh 13.18) A homomorphism is one to one if and only if .

• (Fraleigh 14.11) The Fundamental Homomorphism Theorem. Every Factor Group gives rise to a natural homomorphism.

  Let be a homomorphism with kernel .

  (Fraleigh 13.20) is a normal subgroup subgroup of

  Then is a group and given by is an isomorphism. In other words

  If is onto, then the isomorphism changes to

  If is the homomorphism given by , then

  We refer to and as the natural isomorphism and natural homomorphism respectively.

• An isomorphism is a homomorphism that is bijective.

• An endomorphism is a homomorphism from to itself.

Links §

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