Cyclic Group Group order Abelian group

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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