Section 14 Factor Groups

1. Section 14 Factor Groups Factor Groups from Homomorphisms. Theorem Let  : G G’ be a group homomorphism with kernel H. Then the cosets of H form a factor group, G/H, where (aH)(bH)=(ab)H. Also the map : G/H  [G] defined by (aH)= (a) is an isomorphism. Both coset multiplication and  are well defined, independent of the choices a and b from the cosets.

2. Examples Example: Consider the map : Z  Zn, where (m) is the remainder when m is divided by n in accordance with the division algorithm. We know  is a homomorphism, and Ker () = n Z. By previous theorem, the factor group Z / nZ is isomorphic to Zn. The cosets of n Z (nZ, 1+n Z, …) are the residue classes modulo n. Note: Here is how to compute in a factor group: We can multiply (add) two cosets by choosing any two representative elements, multiplying (adding) them and finding the coset in which the resulting product (sum) lies. Example: in Z/5Z, we can add (2+5Z)+(4+5Z)=1+5Z by adding 2 and 4, finding 6 in 1+5Z, or adding 27 and -16, finding 11 in 1+5Z.

3. Factor Groups from Normal Subgroups Theorem Let H be a subgroup of a group G. Then left coset multiplication is well defined by the equation (aH)(bH)=(abH) If and only if H is a normal subgroup of G.

4. Definition Corollary Let H be a normal subgroup of G. Then the cosets of H form a group G/H under the binary operation (aH)(bH)=(ab)H. Proof. Exercise Definition The group G/H in the proceeding corollary is the factor group (or quotient group) of G by H.

5. Examples Example Since Z is an abelian group, nZ is a normal subgroup. Then we can construct the factor group Z/nZ with no reference to a homomorphism. In fact Z/ nZ is isomorphic to Zn.

6. Theorem Theorem Let H be a normal subgroup of G. Then : G  G/H given by (x)=xH is a homomorphism with kernel H. Proof. Exercise

7. The Fundamental Homomorphism Theorem Theorem (The Fundamental Homomorphism Theorem) Let : G  G’ be a group homomorphism with kernel H. Then [G] is a group, and : G/H  [G] given by (gH)= (g) is an isomorphism. If : G  G/H is the homomorphism given by (g)=gH, then (g)=  (g) for each gG.  G [G]   G/H

8. Example In summary, every homomorphism with domain G gives rise to a factor group G/H, and every factor group G/H gives rise to a homomorphism mapping G into G/H. Homomorphisms and factor groups are closely related. Example: Show that Z4 X Z2 / ({0} X Z2) is isomorphic to Z4.. Note that 1: Z4 X Z2 Z4 by1(x, y)=x is a homomoorphism of Z4 X Z2 onto Z4 with kernel {0} X Z2. By the Fundamental Homomorphism Theorem, Z4 X Z2 / ({0} X Z2) is isomorphic to Z4.

9. Normal Subgroups and Inner Automorphisms Theorem The following are three equivalent conditions for a subgroup H of a group G to be a normal subgroup of G. • ghg-1  H for all g  G and h  H. • ghg-1 = H for all g  G. • gH= Hg for all g  G. Note: Condition (2) of Theorem is often taken as the definition of a normal subgroup H of a group G. Proof. Exercise. Example: Show that every subgroup H of an abelian group G is normal. Note: gh=hg for all h  H and all g  G, so ghg-1 = h  H for all h  H and all g  G.

10. Inner Automorphism Definition An isomorphism : G  G of a group G with itself is an automorphism of G. The automorphism ig: G  G , where Ig(x)=gxg-1 for all xG, is the inner automorphism of G by g. Performing Ig on x is callled conjugation of x by g.