Exercise 3.2. Let G, G' be two groups and : G→G' be a (group) homomorphism. (1) Given a subgroup H of G, show that (H) := {(h) | hЄ H} is a subgroup of G'. (2) Given a subgroup H' of G', show that ¯¹ (H') = {g Є G | ¢(g) Є H'} is a subgroup of G.

12. Find all normal subgroups of the quaternion group.

Show that every subgroup of an abelian group is normal.

Find all homomorphic images of the quaternion group.

Suppose G1 and G2 are groups with normal subgroups H1 and H2, respectively, and with G1/H1 isomorphic to G2/H2. Determine the possible orders of H1 and H2 under the following conditions. a. G1=24 and G2=18 b. G1=32 and G2=40

please solve a question

Theorem 2. Let G, and G, be groups, then @ Gx G, G, x G, (6) If H = {(4, e)| a e G} and H, = {(e, b)| be G, where e, and e, are identity elements of G, and G, respectively, then H, and H, are normal subgroups of G x (ii) H G, and H, G, (iv) The factor (quotient) group (G, xG2)| H; is isomorphic to G2, and (G, × G) H, is isomorphic to G,.

Let the groups: G, G', G'' and the functions f:G-->G', g:G'-->G'' Prove the following statement: If f and g are group isomorphisms the g ○ f (compostion of functions) is also a group isomorphism.

(H,*) is called a of (G,*) if (H,*) is a group.

Q2) Consider the subgroup H = (i) of the the group (C\{0},.). (1) Find H (2) How many element of order 2 in H? explain that H is normal subgroup of (C\{0},.). (3) Show