## Is every group of order 12 Abelian?

We will use semidirect products to describe all groups of order 12 up to isomorphism. There turn out to be 5 such groups: 2 are abelian and 3 are nonabelian. Every group of order 12 is a semidirect product of a group of order 3 and a group of order 4.

### 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.

**How many groups are there in order 12?**

five groups

There are five groups of order 12. We denote the cyclic group of order n by Cn. The abelian groups of order 12 are C12 and C2 × C3 × C2. The non-abelian groups are the dihedral group D6, the alternating group A4 and the dicyclic group Q6.

**How many groups are there of order 12 up to isomorphism?**

So there are two abelian groups of order 12, up to isomorphism, Z2 × Z2 × Z3 and Z4 × Z3.

## Are all groups cyclic?

Every group of prime order is cyclic, because Lagrange’s theorem implies that the cyclic subgroup generated by any of its non-identity elements is the whole group.

### Is R+ A cyclic group?

Proof that (R, +) is not a Cyclic Group.

**What are the 5 groups of order 12?**

gap> SmallGroupsInformation (12); There are 5 groups of order 12. 1 is of type 6.2. 2 is of type c12. 3 is of type A4. 4 is of type D12. 5 is of type 2^2×3. The groups whose order factorises in at most 3 primes have been classified by O. Hoelder.

**Are there simple non abelian groups of this order?**

In particular, there is no simple non-abelian group of this order. (number of abelian groups of order ) times (number of abelian groups of order ) = ( number of unordered integer partitions of 2) times ( number of unordered integer partitions of 1) = .

## How are groups of this order stored in gap?

Here is GAP’s summary information about how it stores groups of this order, accessed using GAP’s SmallGroupsInformation function: gap> SmallGroupsInformation (12); There are 5 groups of order 12. 1 is of type 6.2. 2 is of type c12. 3 is of type A4. 4 is of type D12. 5 is of type 2^2×3.