Yes, every subgroup of a cyclic group is cyclic because every subgroup is a group.
every abelian group is not cyclic. e.g, set of (Q,+) it is an abelian group but not cyclic.
No.
No! Take the quaternion group Q_8.
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.
Four of them.
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).
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.
Yes, every subgroup of a cyclic group is cyclic because every subgroup is a group.
every abelian group is not cyclic. e.g, set of (Q,+) it is an abelian group but not cyclic.
No.
No Q is not cyclic under addition.
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
Let G be the cyclic group generated by x, say. Ten every elt of G is of the form x^a, for some a
No! Take the quaternion group Q_8.
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.
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.