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Let G be the cyclic group generated by x, say. Ten every elt of G is of the form x^a, for some a

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Q: Prove that every a cyclic group is an abelia?
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Every subgroup of a cyclic group is cyclic?

Yes, every subgroup of a cyclic group is cyclic because every subgroup is a group.

Is every abelian group is cyclic or not and why?

every abelian group is not cyclic. e.g, set of (Q,+) it is an abelian group but not cyclic.

Prove that a group of order 5 must be cyclic?

There's a theorem to the effect that every group of prime order is cyclic. Since 5 is prime, the assertion in the question follows from the said theorem.

Is every abelian group is cyclic or not?


Is every finite abelian group is cyclic?

No, for instance the Klein group is finite and abelian but not cyclic. Even more groups can be found having this chariacteristic for instance Z9 x Z9 is abelian but not cyclic

Proof or Disprove 'If every proper subgroup of G is cyclic then G must be cyclic'?

No! Take the quaternion group Q_8.

A cyclic group of length 2 is called identity?

A cyclic group of order two looks like this.It has two elements e and x such that ex = xe = x and e2 = x2 = e.So it is clear how it relates to the identity.In a cyclic group of order 2, every element is its own inverse.

Is every group whose order is less than or equal to 4 a cyclic group?

Yes. The only group of order 1 is the trivial group containing only the identity element. All groups of orders 2 or 3 are cyclic since 2 and 3 are both prime numbers. Therefore, any group of order less than or equal to four must be a cyclic group.

Is it true that an infinite cyclic group may have 3 distinct generators?

A cyclic group, by definition, has only one generator. An example of an infinite cyclic group is the integers with addition. This group is generated by 1.

Is rational number is a cyclic group under addition?

No Q is not cyclic under addition.

Prove that a group of order three is abelian?

By LaGrange's Thm., the order of an element of a group must divide the order of the group. Since 3 is prime, up to isomorphism, the only group of order three is {1,x,x^2} where x^3=1. Note that this is a finite cyclic group. Since all cyclic groups are abelian, because they can be modeled by addition mod an integer, the group of order 3 is abelian.

What is the order of the cyclic group mean?

The order of a cyclic group is the number of distinct elements in the group. It is also the smallest power, k, such that xk = i for all elements x in the group (i is the identity).