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Is it true that an infinite cyclic group may have 3 distinct generators?
Wiki User
∙ 11y ago
An example of an infinite cyclic group is the integers with addition. This group is generated by 1.
Wiki User
∙ 11y ago
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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.
Is every abelian group is cyclic or not?
No.
Proof or Disprove 'If every proper subgroup of G is cyclic then G must be cyclic'?
No! Take the quaternion group Q_8.
In group theory what is a group generator?
In abstract algebra, a generating set of a group Gis a subset S such that every element of G can be expressed as the product of finitely many elements of S and their inverses.More generally, if S is a subset of a group G, then , the subgroup generated by S, is the smallest subgroup of G containing every element of S, meaning the intersection over all subgroups containing the elements of S; equivalently, is the subgroup of all elements of G that can be expressed as the finite product of elements in S and their inverses.If G = , then we say S generatesG; and the elements in S are called generators or group generators. If S is the empty set, then is the trivial group {e}, since we consider the empty product to be the identity.When there is only a single element x in S, is usually written as . In this case, is the cyclic subgroup of the powers of x, a cyclic group, and we say this group is generated by x. Equivalent to saying an element x generates a group is saying that it has order |G|, or that equals the entire group G.My source is linked below.
Let G be a cyclic group of order 8 then how many of the elements of G are generators of this group?
Four of them.
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).
What is finite and infinite cyclic group?
Normally, a cyclic group is defined as a set of numbers generated by repeated use of an operator on a single element which is called the generator and is denoted by g.If the operation is multiplicative then the elements are g0, g1, g2, ...Such a group may be finite or infinite. If for some integer k, gk = g0 then the cyclic group is finite, of order k. If there is no such k, then it is infinite - and is isomorphic to Z(integers) with the operation being addition.
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.
Is every abelian group is cyclic or not?
No.
Is rational number is a cyclic group under addition?
No Q is not cyclic under addition.
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
Prove that every a cyclic group is an abelia?
Let G be the cyclic group generated by x, say. Ten every elt of G is of the form x^a, for some a
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.
