Please provide details of every step while solving these problems

ALGEBRA

HOMEWORK III

Due 23:00 10/07/2022

1. G is a cyclic group of order n and m|n. Prove that there is a unique subgroup

of order m in G.

2. G is a group of order p2 , where p is a prime. Prove that if G has a unique

subgroup of order p, then G is cyclic.

3. If G is a finite group with |G| > 1, and the only subgroups of G are {1} and G,

prove G is a cyclic group with prime order.

4. f : G −→ G′ is a homomorphism, H ′ is a subgroup of G′ , and H = f −1 (H ′ ) =

{g ∈ G|f (g) ∈ H ′ }.

(i). Prove ker f ⊆ H.

(ii). Prove H is a subgroup of G.

(iii). If H ′ is a normal subgroup of G′ , is H a normal subgroup of G? If yes,

prove it; if no, provide a counter example.

5. G is a group. f : G −→ G is defined by f (g) = g 2 . Prove that f is a

homomorphism if and only if G is abelian.

6. Z is the group of integers with addition as composition, and G = {±1} is the

group of ±1 with multiplication.

(i). For a ∈ Z, let fa : Z −→ Z be the function fa (x) = ax for any x ∈ Z. Prove

Aut(Z) = {f1 , f−1 }.

(ii). Let G = {±1} be the group with multiplication. Prove F : Aut(Z) −→ G

defined by F (f ) = f (1) is an isomorphism.

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