# Kyoto2.org

Tricks and tips for everyone

# Are all cyclic groups of the same order isomorphic?

## Are all cyclic groups of the same order isomorphic?

Cyclic groups of the same order are isomorphic. The mapping f:G→G′, defined by f(ar)=br, is isomorphism. Therefore the groups are isomorphic.

Are two groups isomorphic if they have the same order?

Theorem 1: If two groups are isomorphic, they must have the same order. Proof: By definition, two groups are isomorphic if there exist a 1-1 onto mapping ϕ from one group to the other.

### Are two cyclic groups isomorphic?

Two cyclic groups of the same order are isomorphic to each other.

How many cyclic groups are there of order n up to isomorphism?

By the classification of cyclic groups, there is only one group of each order (up to isomorphism): Z/2Z, Z/3Z, Z/5Z, Z/7Z. (the latter is called the “Klein-four group”). Note that these are not isomorphic, since the first is cyclic, while every non-identity element of the Klein-four has order 2.

## Can a cyclic group be isomorphic to a non cyclic group?

The answer to this question claims that these two groups are isomorphic but I believe this is false. Firstly, surely it must be impossible to have a non-cyclic group that is isomorphic to a cyclic one.

Is every cyclic group is Abelian?

All cyclic groups are Abelian, but an Abelian group is not necessarily cyclic. All subgroups of an Abelian group are normal. In an Abelian group, each element is in a conjugacy class by itself, and the character table involves powers of a single element known as a group generator.

### Are all Abelian groups of the same order isomorphic?

Groups posses various properties or features that are preserved in isomorphism. An isomorphism preserves properties like the order of the group, whether the group is abelian or non-abelian, the number of elements of each order, etc. Two groups which differ in any of these properties are not isomorphic.

Can two non cyclic groups be isomorphic?

## What is the order of a cyclic group?

A cyclic group G is a group that can be generated by a single element a , so that every element in G has the form ai for some integer i . We denote the cyclic group of order n by Zn , since the additive group of Zn is a cyclic group of order n .

How many groups of order 4 are there up to isomorphism?

two different groups
There are only two different groups of order 4 up to isomorphism.

### How many abelian groups up to isomorphism are there of order 15?

Table of number of distinct groups of order n

Order n Prime factorization of n Number of Abelian groups ∏ ω (n) i = 1 p (αi)
13 13 1 1
14 2 1 ⋅ 7 1 1
15 3 1 ⋅ 5 1 1
16 2 4 5

Can cyclic group be isomorphic?

Every infinite cyclic group is isomorphic to the additive group of Z, the integers. Every finite cyclic group of order n is isomorphic to the additive group of Z/nZ, the integers modulo n.

## Does isomorphism preserve cyclic?

It is true, the proof amounts to showing that H must be cyclic as a consequence of the operation preserving nature of the isomorphism.

Do all cyclic groups have prime order?

The statement you claim to have contradicted, i.e. that every element of a cyclic group G has order either 1 or |G|, is false.

### What are the properties of cyclic groups?

Properties of Cyclic Group:

• Every cyclic group is also an Abelian group.
• If G is a cyclic group with generator g and order n.
• Every subgroup of a cyclic group is cyclic.
• If G is a finite cyclic group with order n, the order of every element in G divides n.

Can a cyclic and non cyclic group be isomorphic?

## Is group of order 4 always cyclic?

From Group whose Order equals Order of Element is Cyclic, any group with an element of order 4 is cyclic. From Cyclic Groups of Same Order are Isomorphic, no other groups of order 4 which are not isomorphic to C4 can have an element of order 4.

How many abelian groups up to isomorphism are there of order 360?

six different abelian groups
There are six different abelian groups (up to isomorphism) of order 360. A group G is decomposable if it is isomorphic to a direct product of two proper nontrivial subgroups. Otherwise G is indecomposable.

### How many abelian groups up to isomorphism are there of order 16?

14 groups
There are 14 groups of order 16 up to isomorphism.

Does isomorphism preserve order of elements?

Yes. Isomorphisms preserve order. In fact, any homomorphism ϕ will take an element g of order n to an element of order dividing n, by the homomorphism property.