Can a quotient group be cyclic?

Can a quotient group be cyclic?

Let G/Z(G) be the quotient group of G by Z(G). Then G is abelian, so G=Z(G). That is, the group G/Z(G) cannot be a cyclic group which is non-trivial.

What does it mean for a quotient group to be cyclic?

Let H be a normal subgroup of G . Then it can be verified that the cosets of G relative to H form a group. Then |G/Z|=p | G / Z | = p so G/Z is cyclic, thus we may decompose G into the cosets Z,Zg,…,Zgp−1 Z , Z g , . . . , Z g p − 1 for some g∈G g ∈ G . …

How do you know if a quotient group is cyclic?

If H is special, then G/H is cyclic, e.g., if H=G.

What is quotient group in group theory?

A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is “factored” out).

Is quotient group always Abelian?

Every subgroup of an abelian group is normal, and every quotient of an abelian group is abelian. Also, a subgroup of a nonabelian group need not be normal, and a quotient of a nonabelian group need not be abelian.

Is the quotient group of two cyclic groups cyclic?

The quotient of a cyclic group is again cyclic. A cyclic group is a group which is generated by a single element.

Is quotient group always abelian?

How does GH show abelian?

Let G/H denote the quotient group of G by H. Then G/H is abelian if and only if H contains every element of G of the form aba−1b−1 where a,b∈G.

What is a right coset?

Given an element g of G, the left cosets of H in G are the sets obtained by multiplying each element of H by a fixed element g of G (where g is the left factor). The right cosets are defined similarly, except that the element g is now a right factor, that is, Hg = {hg : h an element of H} for g in G.

Is every quotient group of a cyclic group is cyclic?

What is the identity of a quotient group?

Definition: If G is a group and N is a normal subgroup of group G, then the set G|N of all cosets of N in G is a group with respect to the multiplication of cosets. It is called the quotient group or factor group of G by N. The identity element of the quotient group G|N by N.

Is the quotient group Z an abelian subgroup?

Even and odd integers. Consider the group of integers Z (under addition) and the subgroup 2Z consisting of all even integers. This is a normal subgroup, because Z is abelian. There are only two cosets: the set of even integers and the set of odd integers; therefore, the quotient group Z/2Z is the cyclic group with two elements.

How is the quotient group Z / 2 Z isomorphic?

There are only two cosets: the set of even integers and the set of odd integers, and therefore the quotient group Z /2 Z is the cyclic group with two elements. This quotient group is isomorphic with the set {0,1} with addition modulo 2; informally, it is sometimes said that Z /2 Z equals the set {0,1} with addition modulo 2.

How is the cyclic group of addition obtained?

For example, the cyclic group of addition modulo n can be obtained from the group of integers under addition by identifying elements that differ by a multiple of n and defining a group structure that operates on each such class (known as a congruence class) as a single entity. It is part of the mathematical field known as group theory .

Is the quotient of a group always a normal subgroup?

It is part of the mathematical field known as group theory. In a quotient of a group, the equivalence class of the identity element is always a normal subgroup of the original group, and the other equivalence classes are precisely the cosets of that normal subgroup.