Answer to Question #203327 in Abstract Algebra for Raghav

Question

Use FTH to determine all homomorphic images of D₈ upto isomorphism.

Answer 1

Every homomorphic image G of D₈ is isomorphic to D₈/N, where N  is a normal subgroup of D₈.

Therefore, the order of G divides the order of D₈, which is 8. Thus:

• If the order of G is 8, then G $\cong$D₈
• If the order of G is 4, then G $\cong$C₄ or G $\cong$C₂$\times$C₂
• If the order of G is 2, then G $\cong$C₂
• If the order of G is 1, then G $\cong$C₁