 # Abstract Algebra Questions Anonymous
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 7c2d80af_1962_4420_bd82_b6bba7be2981.jpeg dafb0053_3e5b_4c6e_99d7_eb1ccf287cde.jpeg d12f5751_7cd0_4d25_acd4_899900da1a73.jpeg

Borys_S
School: UCLA   The solutions are ready, please ask if something is unclear.

1. Homomorphisms.
𝜑: ℝ → ℝ, 𝜑(𝑥) = 3𝑥 − 5 is NOT a homomorphism.
Indeed, a homomorphism must preserve neutral element, i.e. 𝜑(𝑒) = 𝑒.
For the (additive) group ℝ, the neutral element is 𝑒 = 0.
But 𝜑(0) = −5 ≠ 0. It is simple to check that 𝜑 does not preserve addition, too.

2. Isomorphisms.

𝜑: (ℝ, +) → (ℝ+ ,×), 𝜑(𝑥) = 𝑒 𝑥 .
(a) It is a homomorphism because it preserves neutral element, the group operation and
inversion:
𝜑(𝑒) = 𝜑(0) = 𝑒 0 = 1 = 𝑒 ′ ,
𝜑(𝑥 + 𝑦) = 𝑒 𝑥+𝑦 = 𝑒 𝑥 × 𝑒 𝑦 ,
−1

𝜑(−𝑥) = 𝑒 −𝑥 = (𝑒 𝑥 )−1 = (𝜑(𝑥)) .
(b) It is one-to-one (injective) because if 𝜑(𝑥) = 𝑒 𝑥 = 𝑒 𝑦 = 𝜑(𝑦) implies 𝑥 = 𝑦.
(c) It is onto because for each 𝑦 ∈ ℝ+ there is 𝑥: 𝜑(𝑥) = 𝑦 (we call it 𝑥 = ln 𝑦).
(d) It is an isomorphism by definition: a bijective homomorphism.

3. A Special Normal Subgroup.
...

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Thanks, good work Brown University

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