# Abstract Algebra Questions

*label*Mathematics

*timer*Asked: Feb 21st, 2019

*account_balance_wallet*$15

### Question Description

Hello,

In the attachments three pictures each one has a questions,

Please solve all the three problems,

Thanks for your hard work in advance

## Tutor Answer

The solutions are ready, please ask if something is unclear.

1. Homomorphisms.

π: β β β, π(π₯) = 3π₯ β 5 is NOT a homomorphism.

Indeed, a homomorphism must preserve neutral element, i.e. π(π) = π.

For the (additive) group β, the neutral element is π = 0.

But π(0) = β5 β 0. It is simple to check that π does not preserve addition, too.

2. Isomorphisms.

π: (β, +) β (β+ ,Γ), π(π₯) = π π₯ .

(a) It is a homomorphism because it preserves neutral element, the group operation and

inversion:

π(π) = π(0) = π 0 = 1 = π β² ,

π(π₯ + π¦) = π π₯+π¦ = π π₯ Γ π π¦ ,

β1

π(βπ₯) = π βπ₯ = (π π₯ )β1 = (π(π₯)) .

(b) It is one-to-one (injective) because if π(π₯) = π π₯ = π π¦ = π(π¦) implies π₯ = π¦.

(c) It is onto because for each π¦ β β+ there is π₯: π(π₯) = π¦ (we call it π₯ = ln π¦).

(d) It is an isomorphism by definition: a bijective homomorphism.

3. A Special Normal Subgroup.

...

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