Please do not just copy paste from AI, I need original work. Fundamental homomorphism theorem (FHT) If : G→ H is a homomorphism, then Im(#)G/ Ker(). The FHT says that every homomorphism can be decomposed into two steps: (i) quotient out by the kernel, and then (ii) relabel the nodes via . Let G be a group and let G be the commutator subgroup of G. Prove that the quotient group G/G is abelian and that G' is the smallest normal subgroup of G for which the quotient is abelian. Visualizing the FHT via Cayley graphs G (Ker(6) G) Im(6) H any homomorphism iN quotient process G/Ker(d) group of cosets remaining isomorphism ("relabeling") Qg iN kN "quotient map" N $ = LOTT QB/N jN iN kN VA "relabeling map"

Find two groups of order 6 that are not isomorphic.

32. Let be a fixed element of the group . According to Exercise 20 of section 3.5, the mapping defined by is an automorphism of . Each of these automorphism is called an inner automorphism of . Prove that the set forms a normal subgroup of the group of all automorphism of . Exercise 20 of Section 3.5 20. For each in the group , define a mapping by . Prove that is an automorphism of .

12. Find all normal subgroups of the quaternion group.

For a fixed group G, prove that the set of all automorphisms of G forms a group with respect to mapping composition.

9. Find all homomorphic images of the octic group.

Let G be an abelian group of order 2n, where n is odd. Use Lagranges Theorem to prove that G contains exactly one element of order 2.

Please do not just copy paste from AI, I need original work. Fundamental homomorphism theorem (FHT) If o: G+ H is a homomorphism, then Im(4)G/Ker(). The FHT says that every homomorphism can be decomposed into two steps: (i) quotient out by the kernel, and then (ii) relabel the nodes via . Let G be a group and H a subgroup of G. Show that there exists an exact sequence involving the quotient group G/H and its relation to homomorphisms. Specifically, prove that for a homomorphism : GK where K is a group, the induced map | H→ K can be extended to a homomorphism from G/H to K. Visualizing the FHT via Cayley graphs G Im(6) H (Ker(6) 4G) any homomorphism iN quotient process G/Ke group of cosets remaining isomorphism ("relabeling") Q8 iN KN "quotient map" π N jN = LOT QB/N iN kN VA "relabeling map"

Please do not just copy paste from AI, I need original work. Fundamental homomorphism theorem (FHT) If : G→ H is a homomorphism, then Im(4)G/Ker(). The FHT says that every homomorphism can be decomposed into two steps: (i) quotient out by the kernel, and then (ii) relabel the nodes via . Let G be a group, and let N and M be normal subgroups of G. Prove that if N is isomorphic to M as groups, then G/N is isomorphic to G/M. Hint: Use the Fundamental Homomorphism Theorem to define an isomorphism between the two quotient groups. Visualizing the FHT via Cayley graphs G Im(6) H (Ker(0) 4G) any homomorphism quotient process G/Karld group of cosets remaining isomorphism ("relabeling") iN QU "quotient map" π kN jN $ = LOTT Qo/N iN kN) h VA "relabeling map"

Please do not just copy paste from AI, I need original work. Fundamental homomorphism theorem (FHT) If : G H is a homomorphism, then Im()G/Ker(). The FHT says that every homomorphism can be decomposed into two steps: (i) quotient out by the kernel, and then (ii) relabel the nodes via . Let G be a group, X a set, and pp: G→ Sym(X) a homomorphism, where Sym(X) is the symmetric group on X. Prove that if ker() = {e}, then the map p is injective, and hence G is isomorphic to a subgroup of Sym(X). Hint: Use the Fundamental Homomorphism Theoren and the fact that injectivity of implies that G is isomorphic to its image, which is a subgroup of Sym(X). Proof approach: Visualizing the FHT via Cayley graphs G In(d) SH (Ker()G) any homomorphism quotient process G/Kerle group of cosets remaining isomorphism ("relabeling") QB iN IN kN "quotient map" N = LOT QB/N jN iN KN VA "relabeling map"