Question

If G is an infinite cyclic group then which of the following is not true?

a) G has exactly 2 generators.

b) G is isomorphic to (Z,+).

c) Every proper sub group of  G is finite.

Comments

Every infinite cyclic group is isomorphic to (Z, +)​​​​​​

1. True, G has exactly 2 generators 1 and -1
2. True.
3. False, take counter example as 3z = {3x | x ∈ Z} is a proper subgroup and it is infinite.

Answer 1

ans is A:

because cyclic group is a group generate dby a single element, which is generator g. applying repeatedly group operations to this g or its inverse we can get all the eements of group.

B & C are propertie of infinite cyclic group, which are always true.

Comments

No if x is generator then x^-1 is also generator .So it has to  be either b or c .If it it s c how??

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http://en.wikipedia.org/wiki/Cyclic_group

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i think c is false  not every proper subgroup of G would be finite .correct if i am wrong

Answer 2

All are true:

https://mathoverflow.net/questions/120829/an-infinite-group-such-that-every-proper-subgroup-is-finite
https://math.stackexchange.com/questions/1075889/an-infinite-cyclic-group-has-exactly-two-generators-is-my-proof-correct