# Short-Questions

Fast solutions for complex problems

## Is every group of order less than 4 is cyclic?

The Klein-4 Group is one of two groups of order 4, and it is non-cyclic. However groups of order less than or equal to 3 as well as groups of order 5 are guaranteed to be cyclic. This is because groups of prime order are cyclic and it is easy to prove.

## Why is the group of order 4 always cyclic?

If there’s an element with order 4, we have a cyclic group – which is abelian. Otherwise, all elements ≠e have order 2, hence there are distinct elements a,b,c such that {e,a,b,c}=G.

Are all groups of order 5 cyclic?

. Since 5 is prime, there are no subgroups except the trivial group and the entire group. is therefore a simple group, as are all cyclic graphs of prime order.

### Is every group of order 3 cyclic?

Any group of order 3 must be cyclic The thing that we can prove is that ab = e. Proof: It can not be anything else. If ab = a then b = e , a contadiction. If ab = b then a = e , a contradiction.

### Is every group of order 4 Abelian?

This implies that our assumption that G is not an abelian group ( or G is not commutative ) is wrong. Therefore, we can conclude that every group G of order 4 must be an abelian group. Hence proved.

Is the Klein 4 group Abelian?

One is the cyclic group with four elements. The other is the Klein four-group. It is abelian, and may be written as a multiplicative group with elements e, a, b and c, where e is the identity element, a 2=b 2=c 2=e, ab=c, bc=a and ca=b.

## Is the Klein 4 group a field?

The Klein 4-group is an Abelian group. It is the smallest non-cyclic group. It is the underlying group of the four-element field.

## Are all groups of order 9 cyclic?

Both of these are abelian groups and, in particular are abelian of prime power order….Groups of order 9.

Group GAP ID (second part) Defining feature
cyclic group:Z9 1 unique cyclic group of order 9
elementary abelian group:E9 2 unique elementary abelian group of order 9; also a direct product of two copies of cyclic group:Z3.

Why is group 15 order cyclic?

From Order of Element Divides Order of Finite Group, they are all of order 1, 3, 5 or 15. As the elements of order 1, 3 and 5 have been accounted for, they must all be of order 15. So G has 8 distinct elements of order 15. Hence G must be cyclic.

### Is every group of order 2 cyclic?

The cyclic group of order 2 occurs as a subgroup in many groups. In general, any group of even order contains a cyclic subgroup of order 2 (this follows from Cauchy’s theorem, which is a corollary of Sylow’s theorem, though it can also be proved by a direct counting argument).

### Is S4 cyclic?

Cyclic four-subgroups of symmetric group:S4.

Which of following is group of order 4?

Groups of order 4

Group GAP ID (second part) Defining feature
cyclic group:Z4 1 unique cyclic group of order 4
Klein four-group 2 unique elementary abelian group of order 4; also a direct product of two copies of cyclic group:Z2.

## Is there a cyclic group of order 4?

Moreover, only identity has order equal to 1. So all other elements must have orders 2 or 4. If there is an element of order 4, this group is cyclic. So the only remaining case is that there might be a group with four elements where all non-identity elements are of order two.

## How to prove that every group of order 4 is?

We know that order of any element of a group divides the order of the group. So possible orders of elements of our are 1, 2, 4. Moreover, only identity has order equal to 1. So all other elements must have orders 2 or 4. If there is an element of order 4, this group is cyclic.

Are there any groups of order less than 5?

5 Answers. Any group of prime order is cyclic, hence abelian. This implies that all groups of order 2, 3 and 5 are abelian. On the other hand, all groups of order p2 are abelian. Hence all groups of order 4 are abelian. The trivial group of order 1 is abelian as well. For groups of order 4 the explicit classification also shows the result.

### Which is an isomorphic group of order 4?

Note: actually this group is isomorphic to Klein’s Vierergruppe Z / 2 Z × Z / 2 Z. If G has an element of order 4, then G is cyclic. If G has no element of order 4, then a, b, c are all of order 2. That means a 2 = b 2 = c 2 = e.