# Which group is non-Abelian of order?

## Which group is non-Abelian of order?

A non-Abelian group, also sometimes known as a noncommutative group, is a group some of whose elements do not commute. The simplest non-Abelian group is the dihedral group D3, which is of group order six.

Is every group of order PQ Abelian?

(1) If p does not divide q − 1, then any group G of order pq is cyclic. (2) If p divides q − 1 then there are only two non-isomorphic groups of order pq one of which is commutative (which is again cyclic as p and q are different primes) other is non-commutative. q is Abelian. Theorem 1.3.

Is group of order 21 Abelian?

These must be order 21 and so G is cyclic and hence Abelian. Thus there cannot be a unique group of order 3 and so there are 7 of them. In conclusion the only Sylow subgroup of G is the one of size 7.

### Is there a non-abelian group of order 4?

There is only one other group of order four, up to isomorphism, the cyclic group of order 4. Both are abelian groups. The smallest non-abelian group is the symmetric group of degree 3, which has order 6.

What is abelian and non-abelian?

Definition 0.3: Abelian Group If a group has the property that ab = ba for every pair of elements a and b, we say that the group is Abelian. A group is non-Abelian if there is some pair of elements a and b for which ab = ba.

Is a group of order 18 abelian?

Since 18=2⋅32, the number n3 of Sylow 3-subgroups is 1 by the Sylow theorem. (Sylow’s theorem implies that n3≡1(mod3) and n3 divides 2.) Hence the unique Sylow 3-subgroup P is a normal subgroup of G. The order of P is 9, a square of a prime number, thus P is abelian.

#### Are all groups of order PQ cyclic?

As P and Q are cyclic groups of distinct prime order, P × Q is cyclic and so is G. Therefore, if p q − 1, then all groups of order pq are cyclic.

Is a group of order 15 Abelian?

Group of order 15 is abelian.

How many normal subgroup does a non Abelian group of Order 21?

Hence group of order 21 has atleast one normal subgroup.

## Is a group of order 21 Cyclic?

It has element and each non-identity element has order , hence it is non-cyclic. As it direct product of two abelian groups and hence it is abelian. So it is abelian, non-cyclic group of order .

How many non-Abelian group of order 12 are there?

3 non-abelian groups
We conclude that in addition to the two abelian groups Z12 and Z2 × Z6, there are 3 non-abelian groups of order 12, A4, Dic3 ≃ Q12 and D6.