Past exam of the mathematics course of the University of Cambridge / 2025 / ia / Paper 3 / 2D (Groups)

Table of contents
- i 2D
  - Solution i
- ii 2D
  - Solution ii
- iii 2D
  - Solution iii

i

Solution

False. A finite cyclic group has nonidentity torsion, whereas

Z

is torsion-free, so an injective homomorphism cannot exist.

ii

Solution

False. A surjection

C_{n} \to C_{m}

exists exactly when

m

divides

n

; for example there is none from

C_{3}

to

C_{2}

.

iii

Solution

False. The proper subgroup

2 Z \times 2 Z

of

Z^{2}

is isomorphic to

Z^{2}

and is not cyclic.

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